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COMMON   SCHOOL 


ARITHMETIC 


BY 

D.    B.    HAGAR, 

PRINCIPAL    OF    STATE    NORMAL    SCHOOL,    SALEM,    MASS. 


PHILADELPHIA: 
COWPERTHWAIT   cS:    COMPANY. 


HAGAR'S 

Mathematical  Series. 


I.   Primary  Lessons  in  Numbers. 
II.    Elementary  Arithmetic. 
III.    Common  School  Arithmetic. 


IV.     Elementary  Algebra.    (In  press.) 

To  be  followed  by  other  Books  of  a  Complete  Series. 


FOR    TEACHERS. 

A    Manual    of    Dict.a.tion    Problems    and    Key    to    the 

Common  School  Arithmetic. 

Containing  carefully  prepared  Dictation  Problems  and  Topical  Reviews, 

and    all    answers    omitted   in   the   Common    School    Arithmetic,  with 

Solutions  of   Difficult  Problems         ....         Price,  Si.oo 

Dictation  Problems  (separately)       ....  "50 

Foi-wanied,  postpaid,  on  receipt  of  the  Price. 


Entered.,  accordiitf;  to  Act  of  Coiif^ress,  in  the  year  137 1,  by 

DANIFJ.   E.  HAGAR    and  HENRY  n.  MAGI.ATHLIM 

In  the  Office  of  the  Librarian  of  Congress,  at   Washington. 


Westcott  &  Thohson,  Sherman   &   Co. 

Stereolyptrs  and  EUclrotyfers.  PItilada.  Printers,  Philada, 


INTRODUCTION. 


This  Common  School  Arithmetic  is  designed  to  be  a  com- 
plete manual  for  learners  Avho  may  be  prepared  to  advance 
beyond  the  first  lessons  in  Numbers. 

It  has  been  constructed  with  a  view  to  the  most  rapid  and 
thorough  progress  of  the  pupil  by  the  use  of  the  least  number 
of  books  possible,  and  by  the  greatest  economy  of  time. 

It  combines  mental  and  written  exercises  in  a  practical  sys- 
tem. All  obsolete  and  valueless  material  and  all  merely 
puzzling  problems  have  been  excluded,  but  no  pains  have 
been  spared  to  embody  valuable  modern  methods  of  computa- 
tion and  topics  having  direct  relation  to  business  as  it  is 
transacted  at  the  present  day. 

The  work  is  sufficiently  comprehensive  to  render  the  use  of 
a  higher  arithmetic  quite  unnecessary.  It  is  ample  enough  in 
its  range  of  subjects  and  exercises  to  qualify  the  learner  for  a 
skillful  and  i:)rompt  solution  of  all  ordinary  problems  of  a 
commercial  character,  and  at  the  same  time  to  subserve  the 
purposes  of  mental  discipline. 

The  Primary  Lessons  in  Numbers  and  the  Elementary 
Arithmetic,  of  this  series,  it  is  believed,  form  a  valuable  com- 
•])endious  course  sufficient  for  a  majority  of  pupils.  The 
Primary  Lessons  and  the  Common  School  Arithmetic  like- 
wise form  a  two-book  course,  but  full  and  complete. 


SUGGESTIONS   TO   TEACHEKS. 


A  TEACHER  should  never  undertake  a  recitation  in  Arith- 
metic without  a  full  understanding  of  the  subject  of  the  lesson. 
Preparation  should  be  made  for  the  elucidation  of  difficulties, 
and  for  making  plain  the  way  of  the  learner,  whenever  re- 
quired, by  happily-chosen,  familiar  illustrations. 

In  forming  classes,  pupils  of  the  same  attainments,  as  nearly 
as  possible,  should  be  placed  together.  Vary  the  exercises  in 
a  class,  so  as  to  secure  animation  and  interest  and  retain  the 
attention  of  each  member  during  the  entire  recitation. 

Tolerate  no  indefinite  answering  of  questions.  Require  all 
principles  and  rules  to  be  recited  exactly,  and  all  forms  of  so- 
lution to  be  logically  and  concisely  expressed.  Let  all  answers 
to  Test  Questions  be  definite  and  prompt. 

Do  not  overlook  the  importance  of  mental  arithmetic.  The 
plan  of  this  book  is  to  combine  mental  and  written  exercises, 
and  to  require  a  reason  for  every  process.  The  proficiency 
of  the  learner  should  be  often  tested  with  problems  not  found 
in  the  book.  Require,  as  an  occasional  test,  the  formation  of 
problems  and  their  solution  without  regard  to  rules. 

Seek,  above  all,  to  make  the  arithmetical  exercise  useful  in 
the  cultivation  of  the  invaluable  habit  of  self-reliance.  En- 
deavor to  give  such  a  practical  character  to  the  instruction  in 
the  science  of  Numbers,  that  the  knowledge  acquired  may  be 
found  readily  available  in  the  many  computntions  required  in 
business.  To  attain  (his  object,  good  Ixjoks  are  aids,  but  can 
never  perform  the  duties  or  assume  the  responsibilities  of  the 
'eacher. 
4 


CONTENTS. 


SECTION  PAGiS 

I.  Pkeliminary     Defi- 
nitions   " 

ii.  nubieration  and  no- 
TATION   9 

Orders  of  Units 11 

Decimal  System 12 

III.  Addition 17 

IV.  Subtraction  24 

V.  Review  Problems 30 

VI.  Multiplication 33 

VII.  Division 41 

VIII.  Review  Problems 49 

IX.  Factoring 52 

X.  Divisors    and    Fac- 
tors   56 

Greatest       Common 

Divisors 58 

XI.  Multiples 60 

Least  Common  Mul- 
tiples   62 

XII.  Factors    in     Divis- 
ion   64 

Division  by  Factors...  65 

Cancellation 66 

XIII.  Analysis 68 

XIV.  Review  Problems 70 

XV.  Common  Fractions 72 

XVI.  Reduction   of  Frac- 
tions   75 

Higlier     or     Lower 

Terms 75 

Common  Denomina- 
tor   79 

XVII.  Addition    of    Frac- 
tions   82 

XVIII.  Subtraction  OF  Frac- 
tions   83 

XIX.  Multiplication     of 

Fractions 85 

Compound         Frac- 
tions   88 

XX.  Division     of     Frac- 
tions   90 

Complex  Fractions...  93 
1* 


SECTION  PAQE 

XXI.  Relation    of    Num- 
bers     95 

XXII.  Review  Problems 98 


XXIII.  Decimal  Fractions  102 

XXIV.  Reduction  of  Deci- 

mals   106 

XXV.  Addition  and  Sub- 
traction OF  Dec- 

I3IALS Ill 

XXVI.  Multiplication  of 

Decimals 113 

XXVII.  Division    of    Deci- 
mals    115 

XXVIII.  United  States 

Money 118 

XXIX.  Reduction  of  U.  S. 

Money 120 

XXX.  Computations 122 

XXXI.  Business  Methods...  124 
XXXII.  Bills       and      Ac- 
counts   128 

XXXIII.  Review  Problems...  132 


XXXIV.  Denominate    Num- 
bers   133 

Measures  of  Exten- 
sion   133 

Measures  of  Capa- 
city  139 

Measures  of  Weight  141 
Measures  of  Time,..  144 
Miscellaneous 

Measures 147 

XXXV.  Compound  Numbers  149 
Reduction           De- 
scending   150 

Reduction  Ascend- 
ing   152 

XXXVI.  Addition    of    Com- 
pound Numbers.  157 
XXXVII.  Subtraction         of 
Compound  Nttm- 

bers.... 160 

Difference  of  Dates.  161 
6 


CONTENTS. 


SECTION  PAGE 

XXXVIII.  Multiplication  of 
Compound  Num- 
bers   162 

XXXIX,  Division    of    Com- 
pound Numbers.  161 
Longitude  and 

Time 165 

XL.  Aliquot  Parts 168 

XLI.   MEASURE3IENTS  170 

XLII.  Review  Problems..  174 

XLin.  Percentage 176 

XLIV.  Profit  and  Loss 182 

XLV.  Commission 185 

XLVI.  Insurance 187 

XLVII.  Review  Problems..  189 

XLVIII.  Simple  Interest 192 

XLIX.  Partial  Payments.  202 

L.  Present  Worth 206 

Commercial       Dis- 

couiit 206 

True  Discount 207 

LI.  Banking -  208 

LII.  Annual  Interest...  213 
LIII.  Compound     Inter- 
est    215 

LIV.  Review  Problems..  217 

LV.  Ratio 219 

LVI.  Proportion 221 

Simple  Proportion.  223 
Compound         Pro- 
portion  225 

LVII.  Distributive  Pro- 
portion   229 

LVIII.  Partnership 2.30 

LIX.  Average    of  Pay- 
ments  235 

Debit    and    Credit 

Account 239 

Cash  Balances 241 

LX.  Investments 212 

Corporate  Stocks 213 

Government  Secur- 
ities  243 

LXI.  Exchange 247 

Domestic            Ex- 
change    248 

Foreign  Exchange..  250 

LXII.  General  Taxes 254 

LXIIl.  National  TAxes....  257 
LXIV.  Review  Problems.  258 


SECTION  PAGB 

LXV.  Involution 262 

Powers 263 

Process     of    Involu- 
tion  263 

LXVI.  Evolution 264 

INlethod    for    Square 

Root 265 

Method      for      Cube 
Root 271 

LXVII.  Mensuration 278 

Polygons 278 

Right  -  angled       Tri- 
angles  282 

Circles 283 

Volumes 284 

Similar  Figures 287 

LXVIII.  Boards  and  Timber...  290 

Squared  Timber 290 

Round  Timber 291 

LXIX.  Stone      and      Brick 

Work 293 

LXX.  Grain  and  Hay 295 

LXXI.  Gauging 296 

LXXII.  Metric  Sy'Stem 298 

Linear  Measures 299 

Surface  Measures 299 

Cubic  Measui-es 299 

Liquid  and  Dry  Meas- 
ures   300 

Weights 300 

Computations 302 

LXXIII.  Series    or    Progres- 
sion  304 

Arithmetical       Pro- 
gression  301 

Geometrical  Progres- 
sion  306 

LXXIV.  Life  Interests 308 

Carlisle  Tables 309 

LXXV.  Problems  for  Anal- 
ysis  311 

LXXVI.  General  Review 321 

APPENDIX. 

Roman  Notation 323 

Contractions 326 

Duodecimals. 328 

Accurate  Interest 3;i0 

Examination  Problems 331 


Common  School  Arithmetic. 


SECTION   I. 

PRELIMIJfAET  BEFIKITIOKS. 

'RTICLE  1. — A  Unit  is  one,  or  a  single  thing  of  any 
kind. 
3.  A  Number  is  a  unit,  or  a  collection  of  units. 
Thus,  one,  two,  three,  four,  five,  are  numbers. 

3.  The  Unit  of  a  number  is  one  of  the  collection  forming 
that  number. 

Thus,  one  is  the  unit  of  six,  one  book  is  tlie  unit  of  six  books. 

4.  An  Integer  is  a  number  formed  wholly  of  entire  units. 

Thus,  three,  five,  six,  nine,  are   integers.     Integers  are  also  called 
integral  or  vohoh  numbers. 

6,  Similar  Numbers  are  those  which  have  the  same  unit. 

Thus,  three  yards  and  five  yards  are  similar  numbers. 

6.  Dissimilar  Numbers  are  those  which  do  not  have  the  same 
unit. 

Thus,  three  yards  and  three  books  are  dissimilar  numbers. 

7.  A  Concrete  Number  is  one  that  names  the  kind  of  unit 
numbered. 

Thus,  five  bushels,  in  which  the  kind  of  unit  is  named,  is  a  concrete 
number. 

7 


8  PRELIMINARY  DEFINITIONS. 

8.  An  Abstract  Number  is  one  that  does  not  name  the  kind 
of  unit  numbered. 

Thus,  five,  in  which  the  kind  of  unit  is  not  named,  is  an  abstract 
number. 

9.  Arithmetic  is  the  science  of  numbers  and  the  art  of  com- 
puting by  them. 

10.  A  Solution  in  Arithmetic  is  the  process  of  answering  a 
question  which  requires  computation. 

11.  A  Proof  of  a  solution  is  the  process  of  testing  its  cor- 
rectness. 

12.  A  Problem  is  a  question  for  solution. 

13.  A  Principle  is  a  general  truth. 

14.  A  Rule  is  U  concise  statement  of  the  method  of  solving 
a  problem. 

15.  An  Example  is  a  problem  which  is  used  to  illustrate  a 
principle  or  rule. 

16.  An  Exercise  is  a  problem  which  is  intended  to  render 
knowledge  familiar  by  drill  or  practice. 

EXEItCISES. 

1.  How  many  units  in  one?    In  one  dollar?    Three  is  a  col- 
lection of  how  many  units  ? 

2.  What  is   the   unit  of  two  books?      Of  four?     Of  five 
pounds  ?     Of  seven  houses  ? 

3.  Are  two  cents  and  five  cents  similar  or  dissimilar  numbers  ? 
Why  are  three  men  and  five  books  dissimilar  numbers  ? 

4.  Is  four  yards  a  concrete  or  an  abstract  number  ?     Three  ? 
Two  boys  ? 

5.  Why  is  one  mile  the  unit  of  four  miles  ?     Why  is  one  the 
unit  of  six  ? 

6.  Why  is  two  houses  a  concrete  number  ?     Why  is  four  an 
abstract  number  ? 

7.  Of  the  two  numbers,  two  miles  and  ten  miles,  what  is  the 
unit  ? 

8.  Of  the  two  numbers,  seven  dollars  and  nine  dollars,  what 
is  the  unit? 


NUMERATION  AND  NOTATION. 


SECTION  11. 
JfVMEEATIOK  AJfD  JTOTATIOJ^. 

17.  The  Naming  of  numbers  is  performed  by  means  of  a 
small  number  of  words. 

A  single  thing  is  named  one;  one  and  one  is  named  two; 
one  and  one  and  one  is  named  three;  and  so  we  have  the 
separate  names, 

One,  two,  three,  four,  five,  six,  seven,  eight,  nine,  ten. 

18.  Ten,  by  being  regarded  as  forming  a  set  or  collection  of 
units,  may  be  treated  as  a  single  thing,  or  as  a  unit  equal  to 
ten  ones. 

One  and  ten,  two  and  ten,  three  and  ten,  four  and  ten,  etc., 
by  change  of  form,  give  the  familiar  names, 

Eleven,  twelve,  thirteen,  fourteen,  fifteen,  sixteen,  seventeen, 
eighteen,  nineteen. 

Two  tens,  three  tens,  four  tens,  etc.,  by  change  of  form,  give 
the  names, 

Twenty,  thirty,  forty,  fifty,  sixty,  seventy,  eighty,  ninety. 

Twenty  and  one,  twenty  and  two,  etc. ;  thirty  and  one,  thirty 
and  two,  etc.,  to  ninety  and  nine,  by  change  of  form,  give  the 
names, 

Twenty-one,  twentij-two,  etc.;  thirty-one,  thirty-two,  etc.,  to 
ninety-nine. 

19.  One  Hundred  is  the  name  given  to  a  collection  of  ten  tens. 
One  hundred  and  one  hundred,  two  hundred  and  one  hun- 
dred, etc.,  form 

Two  hundred,  three  hundred,  etc.,  to  nine  hundred. 

FIGURES. 

20.  Fi.g-ures  are  the  characters  commonly  used  to  represent 
numbers.     They  are  as  follows — 

PRINTED,        0,       1,     2,      3,       4,       5,       6,       7,       8,      9. 

WRITTEN,         ^/fJ-Z^^/cP/ 
NAMED,  Zero,      one,     two,     three,     four,       five,        six,      seven,    eight,    nine. 


10  NUMERATION   AND    NOTATION. 

The  figure  0  is  sometimes  called  a  Cipher,  or  Naught,  be- 
cause when  written  alone  it  expresses  no  value,  or  the  absence 
of  number;  and  the  figures  1,  2,  3,  4,  5,  6,  7,  8,  9  are  called 
Numerals,  or  Significant  Figures,  because  each  expresses  as 
many  ones  as  are  denoted  by  its  name. 

Numbers  greater  than  nine  are  expressed  by  repeating  or 
combining  two  or  more  of  the  ten  figures. 

21.  Exact  Tens  are  written  with  the  figure  expressing  the 
number  of  tens  at  the  left  of  0,  which  marks  the  absence  of 
ones ;  and  tens  and  ones  are  wi'itten  with  the  figure  expressing 
the  tens  at  the  left  of  the  figure  expressing  the  ones.     Thus, 

Ten,  or  1  ten. 

Eleven,  or  1  ten  and  1  one. 

Twelve,  or  1  ten  and  2  ones. 

Thirteen,  or  1  ten  and  3  ones. 

Twenty,  or  2  tens. 

Twenty-one,  or  2  tens  and  1  one. 

Thirty,  or  3  tens, 

and  so  on  to  ninety-nine,  or  9  tens  and  9  ones. 

22.  Exact  Hundreds  are  written  with  the  figure  expressing 
the  hundreds  at  the  left  of  two  zeros. 

Hundreds,  tens  and  ones  are  expressed  together  in  a  number 
by  writing  the  figure  expressing  the  tens  at  the  left  of  the 
figure  expressing  the  ones,  and  the  figure  expressing  the  hun- 
dreds at  the  left  of  that  expressing  the  tens.     Thus, 

One  hundred,  or  1  hundred  0  tens  0  ones,  is  written  100. 
Two  hundred,  or  2  hundreds  0  tens  0  ones,  is  written  200. 
Four  hundred  ten,  or  4  hundreds  1  ten  0  ones,  is  written  ^10. 
Five  hundred  six,  or  5  hundreds  0  tens  6  ones,  is  written  506. 
Nine  hundred  seventy-eight,  or  9  hundreds  7  tens  8  ones, 
is  written  978. 

23.  Numeration  is  the  method  of  naming  numbers,  and  of 
reading  numbers  expressed  by  figures. 

24.  Notation  is  the  method  of  writing  numbers,  or  of  express- 
ing numbers  by  fitnires. 


is  written  10, 

ii 

11, 

(( 

12, 

li 

13, 

11 

20, 

(( 

21, 

« 

30, 

NUMERATION   AND    NOTATION.  U 

WRITIEN  EXERCISES. 

Write  in  figures — 

1.  One  hundred  thirty-six;  two  hundred  thirteen. 

2.  Four  hundred  forty-four  ;  one  hundred  eleven. 

3.  Three  hundred  twenty-five ;  five  hundred  ten. 

4.  Six  hundred  seventeen  ;  two  hundred  twenty. 

5.  Seven  hundred  five ;  eight  hundred  fifteen. 

6.  Nine  hundred  nine  ;  seven  hundred  four. 

7.  Five  hundred  ;  eight  hundred  seventy-one. 

8.  Six  hundred  ;  three  hundred  eighty. 

9.  Five  hundred  twenty-two  ;  nine  hundred  ninety-nine. 
10.  Seven  hundred  one ;  three  hundred  twenty-five. 

ORDERS  AND  PERIODS  OF  UNITS. 

25.  Orders  of  Units  are  denoted  by  the  successive  figures 
used  in  expressing  a  number. 

Thus,  in  365,  the  5,  which  expresses  5  ones,  denotes  units  of  the  First 
Order;  the  6,  which  expresses  6  tens,  denotes  units  of  the  Second  Order; 
and  the  3,  which  expresses  3  hundreds,  denotes  units  of  the  Third 
Order. 

26.  In  naming  numbers,  the  first  three  orders  of  units  are 
regarded  as  forming  a  group,  called  the  Class,  or  Period,  of 
Units,  having  ones,  tens  and  hundreds. 

Thus,  425  forms  a  period  composed  of  425  units. 

27.  Ten  hundreds  form  One  Thousand;  ten  thousands  form 
One  Ten-Thousand;  and  ten  ten-thousands  form  One  Hundred- 
Thousand. 

These  three  orders  of  units  form  a  group,  called  the  Period 
of  Thousands,  having  ones,  tens  and  hundreds. 

Thus,  363425  is  composed  of  363  thousands  425  units,  or  of  two  periods, 
and  is  read  tliree  hundred  sixty-three  thousands  four  hundred  twenty- 
five. 

406007  is  composed  of  406  thousands  007  units,  or  of  two  periods,  and 
is  read  four  hundred  six  thousands  seven. 

In  like  manner  are  formed  and  read  other  periods. 


12 


NVMERATION  AND  NOTATION. 


THE   DECIMAL  SYSTEM. 

28.  Simple  or  Primary  Units  are  the  units  expressed  by  a 
numeral  when  written  alone,  or,  by  the  order  of  ones  in  the 
period  of  units,  when  in  a  collection. 

In  writing  numbers,  the  order  of  simple  units  may  be  marked 
by  placing  a  point  (.),  called  the  Decimal  Point,  at  the  right 
of  the  units'  period ;  and  the  different  periods  may  be  separated 
by  a  Comma  (,). 

Thus,  215  thousands  463  units  may  be  written,  215,463. 

Units  are  understood  to  be  Primary  Units  when  not  other- 
wise indicated  by  the  expression  or  its  connection. 

29.  The  Names  of  the  Orders  of  Units,  and  the  Names  of  the 
Periods,  are  given  in  the  following 


TABLE. 


6th 
Period. 


5th 
Period. 


4th 
Period. 


3d 

Period. 


2d 

Period. 


1st 
Period. 


'~t^ 

.— ^ 

,— -s 

.^— V 

^-— V 

'^"^ 

f        1 

•S 

NAMES    OF 

J       3 

a 

a 

o 

s 

!3 

PERIODS. 

o 

a 

o 

•3 

§ 

i 

3 

u 

•^ 

J3 

3 

V        G? 

tH 

n 

§ 

EH 

t3 

(M 

^ 

^ 

«H 

<« 

«M 

O 

O 

o 

o 

o 

o 

ORDERS    OF 
UNITS. 


W   t<  O         H   E^  O 


Weho       Who        WtHO       Weho 


420,    73  5,     80  0,     612,     309,     25  4. 


The  number  expressed  is  Four  hundred  hceniy  quadrillions, 
seven  hundred  thirty-Jive  trillions,  eight  hundred  billions, 
six  hundred  twelve  iniillions,  three  hundred  nine  thousands, 
tivo  hundred  Jiftij-four  units. 

In  reading  numbers  the  name  of  the  units'  period  is  not 
usually  given,  since  when  omitted  it  is  readily  understood. 


NUMERATION  AND   NOTATION.  18 

30.  The  Periods  above  Quadrillious,  in  order,  are — 
Quiniillions,  SexUllions,  Septillions,  Octillions,  Nonillions,  De- 

cillions,  Undecillions,  Duodecillions,  I'redecillions,  Quatuordecil- 
lions,  QuindecUlions,  SexdecilUons,  Septendecillions,  Octodecil- 
lions,  Novendecillions,  Vigintillions,  etc. 

31.  The  Scale  of  numbers  is  the  arrangement  of  their  units. 
In  the  ordinary  system  of  notation,  or  that  which  has  been 

explained,  the  scale  is  ten,  because  the  units  are  so  arranged 
that  ten  ones  are  one  ten,  ten  tens  are  one  hundred,  ten  hundreds 
are  one  thousand,  etc. 

32.  The  method  of  expressing  numbers  by  ten  figures  is 
termed  the  Arabic  Notation,  from  its  having  been  introduced 
into  Europe  by  the  Arabs. 

33.  The  method  of  expressing  numbers  by  the  scale  of  ten 
is  termed  the  Decimal  System,  from  the  Latin  decern,  which  sig- 
nifies ten;  and  the  scale  often  is,  for  the  same  reason,  termed  the 
Decimal  Scale. 

Principles  of  Numeration  and  Notation. 

34. — 1.  Ten  ^(n^ts  of  any  order  in  a  number  are  ahvays  equal 
to  one  unit  of  the  next  higher  order. 

2.  The  same  figure  represents  invariably  the  same  number  of 
units. 

3.  The  name  and  value  of  the  units  represented  by  a  figure  in 
a  number  are  always  those  of  its  order  in  that  number. 

4.  The  absence  of  units  in  any  order  in  a  number  is  marJced 
by  a  cipher. 

5.  The  order  of  simple  units  in  a  number  may  be  known  by 
having  the  decimal  point  expressed  or  understood  at  the  right 
of  that  order. 

EXMUCISES   IN  NUMERATION. 

35.  --Ex.  1.  Write  and  read  56073402. 

Solution.— 56073402,  separated  into  periods,  is  56,073,402,  or  56  mil- 
lions 73  thousands  402  units,  and  is  read  fifty-six  millions  seventy-three 
thousands  four  hundred  two. 

2.  Write  and  read  735467005.     Write  and  read  93606121. 
2 


14 


NUMERATION  AND  NOTATION. 


36.  Rule  for  Numeration.  — Beginning  with  the  lowest 
order  of  units,  separate  the  figures  of  the  number  into 
periods  of  three  figures  each. 

In  reading  begin  at  the  left;  read  the  hundreds ,  tens 
and  ones  of  each  period,  and  give  the  name  of  each 
period,  except  the  last,  after  its  ones. 


Write  and  read- 

1.  3U. 

2.  1780. 
Z.  23U- 

4.  16110. 

5.  70008. 

6.  134-020. 

7.  68110. 

8.  89000. 

9.  143211. 

10.  456104. 

11.  215779. 


FJtOBLJEMS. 

12.  132401. 

13.  3000835. 

14.  92416512. 
16.  732534902. 

16.  7324768291. 

17.  44444444444. 

18.  56073014211597. 

19.  313134405678012. 

20.  14132486879011326. 

21.  59444632132007955. 

22.  3567890038531900210. 


EXERCISES   IN  NOTATION. 

37. — Ex.  1.  Write  in  figures  twenty-two  millions  four  hun- 
dred six  thousands. 

Solution. — Writing  22  for  tlie  tens  and  ones  of 
22,406,000        millions,   40G   for   the   hundreds,   tens   and   ones   of 
thousands,  000  for  the  absence  of  hundreds,  tens  and 
ones  of  units,  gives  22,406,000  as  the  required  expression. 

2.  Write  in  figures  three  hundred  sixty-five  millions  nine 
hundred  twenty-five  thousands  seven  hundred  seventy-five. 

8.  Write  in  figures  nine  hundred  thirty-two  thousands  four 
hundrefl  forty-seven. 

4.  Write  in  figures  four  hundred  eighteen  millions  eight 
hundred  sixty-three  thousands  two  hundred  three. 


NUMERATION  AXD   NOTATION.  16 

38.  Rule  for  Notation.—  Wi'ite  the  figures  representing  the 
hundreds,  tens  and  ones  of  each  period  uv  their  order. 

Mark  hij  a  cipher  any  order  in  the  number  which 
has  no  units  given. 

PROBLJEMS. 

Write  in  figures — 

1.  Three  hundred  fourteen  ;  four  hundred  ten. 

2.  Five  hundred  six  ;  nine  hundred  seventy-seven. 

3.  Sixteen  thousand  ninety-one ;  twenty -five  thousand  one 
hundred. 

4.  One  hundred  eighty-three  thousand ;  two  hundred  nine 
thousand  ninety-nine. 

5.  Nine  thousands  seven  hundreds  three  tens  four  ones. 

6.  Four  millions  six  ;  ten  millions  ;  five  hundred  five  millions, 

7.  Five  hundred  thousands  four  hundred  six ;  one  hundred 
one  thousands  one  hundred  one. 

8.  Thirty-seven  millions  one  hundred  seventy-one  thousands 
eleven. 

9.  Two  hundred  forty-nine  millions  ;  seventeen  billions  nine 
millions. 

10.  Ninety-three  thousands  one  hundred  eighty-six. 

11.  One  hundred  fifty-two  .millions  four  hundred  twenty-five 
thousands  three  hundred  thirty-three. 

12.  Seven  hundred  fifty -five  trillions  one  hundred  six  bil- 
lions four  hundred  fifteen  millions  one  hundred  five  units. 

13.  One  quintillion  twenty-five  quadrillions  one  hundred  fif- 
teen trillions  seven  billions  eight  hundred  eighty-eight  millions 
five  hundred  fifty  units. 

14.  Eight  hundred  eighty-eight  quintillions  six  thousand  six 
hundred  six. 

15.  Three  hundred  thirty-seven  billions  four  hundred  forty- 
nine  millions  two  thousands  three  hundred  eleven. 

16.  Five  decillions  one  hundred  six  nonillions  eight  octil- 
lions four  septillions  one  hundred  nineteen  sextiilions  six  hun- 
dred seventy-nine  quintillions  four  quadrillions  three  hundred 
fifteen  trillions  seven  hundred  twenty  billions  forty-six  millions 
three  thousands  one. 


16  NUMERATION  AND  NOTATION. 

TEST    QUESTIONS. 

39. — 1.  "What  is  a  Unit  ?  A  number  ?  Name  some  number.  What 
is  the  unit  of  a  number  ?  Give  an  illustration  of  the  unit  of  a  number. 
What  is  an  integer  ? 

2.  What  are  Similar  Numbers?  Name  two.  AVhat  are  dissimilar 
numbers  ?  Name  two.  What  is  a  concrete  number  ?  What  is  an  ab- 
stract number  ?  Give  an  illustration  of  a  concrete  number.  Of  an  ab- 
stract number. 

3.  What  is  Arithmetic?  A  solution?  A  proof  ?  A  problem?  A 
principle  ?     A  rule  ?     An  example  ?     An  exercise  ? 

4.  How  is  the  Naming  of  numbers  performed  ?  Give  the  names  of 
the  first  ten  numbers.  How  may  ten  be  regarded  ?  What  numbers  do 
we  get  by  combining  ten  with  each  of  the  first  nine  numbers,  and  chang- 
ing their  form?  Two  tens,  three  tens,  etc.,  by  change  of  form  give  what 
numbers  ?     Twenty  and  one,  twenty  and  two,  etc.  ? 

5.  What  are  Figures  ?  What  does  0  written  alone  express  ?  What 
is  0  called?  What  does  each  of  the  other  nine  figures  express  when 
written  alone?  What  are  they  called?  How  are  exact  tens  written? 
How  are  the  numbers  between  the  tens  written  ?  How  are  exact  hun- 
dreds written  ?  How  are  hundreds,  tens  and  ones  expressed  together  in 
a  number? 

6.  What  is  Numeration?  What  is  notation?  How  are  orders  of 
units  denoted  in  expressing  a  number? 

7.  What  are  Simple  Units,  or  units  of  the  first  order  ?  Units  of  the 
second  order?  Units  of  the  third  order?  How  many  simple  units 
does  2  of  tlie  first  order  express  ?  2  of  the  second  order  ?  2  of  the  third 
order  ? 

8.  How  many  orders  compose  a  Period  ?  What  is  the  name  of  the 
first  period  ?  Of  the  second  ?  Name  the  orders  of  each.  How  many 
units  in  the  period  of  units  equal  one  unit  in  the  period  of  thousands? 
Give  tlie  names  of  the  first  six  periods  in  their  order.  Name  in  order 
])erio(ls  higher  than  the  period  of  quadrillions.  How  may  the  order 
of  simple  units  be  marked?  How  may  tlie  diflerent  periods  be  sepa- 
rated ? 

9.  What  is  the  Scale  of  numbers?  Wiiat  is  the  scale  in  the  ordinary 
system?  Wliat  is  the  method  of  expressing  numbers  by  figures  called? 
Of  expressing  nimibcrs  by  tlie  scale  of  ten?  What  is  the  scale  of  ten 
termed  ? 

10.  Recite  the  Principles  of  numeration  and  notation.  The  rule  for 
numeration.     The  rule  for  notation. 


ADDITION.  17 

SECTION   III. 

ADDITION. 

40. — Ex.  1.  If  you  have  5  dollars  and  your  brother  has  4, 
how  many  dollars  have  both  ? 

2.  If  James   has  6  books  and  John  has  7,  how  many  have 
both  ? 

3.  How  many  cents  are  9  cents  and  3  cents  ? 

4.  Add  by  2's  from  1  to  21. 
Solution.— 1,  3,  5,  7,  9,  11,  13,  15,  17,  19,  21. 

5.  Add  by  3's  from  1  to  22.  Add  by  3's  from  22  to  52. 

6.  Add  by  4's  from  2  to  34.  Add  by  4's  from- 34  to  62. 

7.  Add  by  5's  from  3  to  33.  Add  by  5's  from  33  to  68. 

8.  Add  by  6's  from  4  to  34.  Add  by  6's  from  34  to  70. 

9.  Add  by  7's  from  5  to  40.  Add  by  7's  from  40  to  75. 

10.  Add  by  8's  from  6  to  46.     Add  by  8's  from  46  to  94. 

11.  Add  by  9's  from  7  to  52.     Add  by  9's  from.  52  to  88. 

12.  What  is  the  sum  of  8  dollars  and  7  dollars  ? 

13.  What  is  the  unit  of  8  dollars  and  7  dollars  ? 

14.  Why  are  8  dollars  and  7  dollars  similar  numbers  ?    What 
is  the  unit  of  their  sum  ? 

15.  Are  9  books  and  8  dollars  similar  or  dissimilar  numbers? 

16.  Why  cannot  9  books  and  8  dollars  be  united  into  one 
number  ? 

Because  9  books  and  8  dollars  are  neither  17  books  nor  17  dollars. 

17.  Only  what  kind  of  numbers,  then,  can  be  united  so  as  to 
form  one  number  ? 

18.  In  my  garden  there  are  5  roses  upon  one  bush,  7  upon 
another  and  2  upon  another.     How  many  roses  are  there  in  all  ? 

19.  How  many  ones  are  5,  7  and  2?     7,  5  and  2?     7,  2 
and  5  ?     2,  7  and  5  ?     2,  5  and  7  ? 

20.  When  the  same  numbers  are  added  in  different  orders, 
is  the  result  changed  ? 

21.  4,  3,  5,  1  and  2  are  how  many  ? 

22.  8,  1,  9,  3,  2  and  5  are  what  number? 


IS  ADDITION. 

DEFINITIONS. 

4-1.  Addition  is  the  process  of  uniting  two  or  more  numbers 
to  find  their  sum. 

42.  The  Sum  is  the  result  of  the  addition.  It  contains  as 
many  ones  as  all  the  numbers  added. 

43.  A  Si^  is  a  mark  used  for  abbreviating  an  expression. 

44.  The  Sign  of  Addition  is  +>  and  is  called  plus.  When 
placed  between  two  numbers,  it  means  that  they  are  to  be  added. 

45.  The  Sign  of  Equality  is  =,  and  is  read  equals  or 
equal  When  placed  between  arithmetical  expressions  it  de- 
notes that  they  are  equal. 

Thus,  8  +  5  =  13,  is  read,  eight  plus  five  equaU  thirteen,  and  means  that 
the  sum  of  eight  and  five  is  thirteen. 

Principles  of  Addition. 

46. — 1.    Only  similar  numbers  can  he  added. 

2.  The  sum  and  the  numbers  added  must  be  similar. 

3.  The  sum  of  numbers  will  be  the  same  in  whatever  order 
they  are  added. 

CJ^STS   I. 

When  the  Sum  of  all  the  Units  of  each  Order  is  Less  than  Ten. 

47.— Ex.  1.  What  is  the  sum  of  3142,  2320  and  516? 

r  SI42  Solution. — For  convenience  in  adding,  write 

JS,  umbers  j  ^g^O        ^^^^  numbers  so  that  figures  representing  units  of 
added,      J       _  ^  „        the  same  order  stand  in  the  same  cohimn. 

^ Add  the  ones,  tens,  hundreds,  and  thousands 

Sum,      5978        separately. 
6  ones  +  0  ones  +  2  ones  are  8  ones,  which  write   for  the  ones   of 
the  sum. 

1  ten  -f-  2  tens  +  4  tens  are  7  tens,  wliich  write  for  the  tens  of  the 

sum. 

T)  hundreds  +  3  hundreds  +  1  liundrcd  are  9  iiundreds,  wliich  write 
for  the  hundreds  of  the  sum. 

2  thousands  +  3  thousands  are  5  thousands,  which  write  for  the  thou- 
sands of  the  sum. 

Hence,  the  sum  is  5  thousands  9  hundreds  7  tens  8  ones,  or  5978. 


ADDITION. 

■ite  and  add — 

(2.) 

61 

25 

(3.) 

187 
12 

(4.) 

42 
55 

(5.) 

423 
354 

19 


6.  What  is  the  sum  of  13,  173  and  202?  Ans.  388. 

7.  How  many  dollars  are  101  dollars,  65  dollars  and  113 
dollars  ? 

8.  How  many  books  are  341  books,  113  books  and  202 
books?  Ans.  656. 

C^SE    II. 

Wlien  the  Suiu  of  all  the  Units  of  auy  Order  is  Greater  than  Ten. 

48.— Ex.  1.  What  is  the  sum  of  5591,  1428  and  2335? 

/Solution. — Write  the  numbers  as  in  the  pre- 
OOJl        ceding  case. 
,  ,    ,      X    1428  Begin  to  add  with  the  ones,  so  that  wlien  the 

'      I  2335        sura  of  any  of  the  orders  of  units  is  greater  than 

r»  ~r\(r>r  i        nine,  its  tens  may  be  conveniently  added  with 

bum,      y3o4.      ^,        -^     i- ,,      ^  1  •  1  1 

the  units  of  the  next  liigher  order. 

5  ones  +  8  ones  +  1  one  are  14  ones,  or  1  ten  and  4  ones.  Write  tlie 
4  ones  for  the  ones  of  tlie  sum,  and  reserve  the  1  ten  to  add  with  the  tens. 

1  ten  +  3  tens  -f  2  tens  +  9  tens  are  15  tens,  or  1  hundred,  and  0  tens. 
Write  the  5  tens  for  the  tens  of  the  sura,  and  reserve  the  1  hundred  to 
add  with  the  hundreds.  ' 

1  hundred  +  3  hundreds  +  4  hundreds  +  5  hundreds  are  13  hundreds, 
or  1  tliousand  and  3  hundreds.  Write  the  3  lumdreds  for  the  liun- 
dreds  of  the  sum,  and  reserve  the  1  thousand  to  add  witli  the  thousands. 

1  thousand  +  2  thousands  +  1  thousand  +  5  thousands  are  9  thou- 
sands, wliich  we  write  for  the  thousands  of  the  sum. 

Hence,  the  sum  is  9  thousands  3  hundreds  5  tens  4  ones,  or  9354. 

The  explanation  may  be  abbreviated  by  naming  only  results.     Thus, 

Five,  thirteen,  fourteen  ;  write  4  and  reserve  1.  One,  four,  six,  fifteen ; 
write  5  and  reserve  1.  One,  four,  eight,  thirteen;  write  3  and  reserve  1. 
One,  three,  four,  nine;  write  9.  Ans.  9354. 

Proof. — Tlie  correctness  of  the  sohition  may  be  proved  by  reviewing 
the  work  carefully,  or  by  adding  the  numbers  downward.  If  the  work 
is  correct,  the  result  in  each  case  v/ill  equal  the  first  result,  since  the 
sum  of  numbers  must  be  the  same  in  whatever  order  they  may  be 
added.  (Art.  46—3.) 


20 


ADDITION. 

Add  and  prove — 

C-^-) 

(3.) 

(4.) 

(5.) 

(6.) 

1683 

467 

1062 

114 

7703 

456 

305 

3457 

590 

4104 

312 

587 

2004 

309 

3492 

7.  What  is  the  sum  of  615  +  3045  +  5000  ? 

8.  How  many  are  3600,  7240  and  797  ? 

9.  A  farmer  raised  from  one  field  1213  bushels  of  wheat ; 
from  a  second,  1308  bushels  ;  from  a  third,  2230  bushels  ;  and 
from  a  fourth,  244  bushels.     How  much  wheat  did  he  raise  ? 

Ans.  4995  bushels. 

40.  Rule  for  Addition.— TFrJ?5e  tlte  nimvhei^s  so  that  all 
figures  of  the  same  order  shall  standi  in  the  same 
column,  and  draiv  a  line  under  them. 

Begin  at  the  right,  add  the  units  of  each  order  sepa- 
rately, and  lurite  the  sum,  if  less  than  ten,  under  the 
column  added. 

If  the  sum  is  ten  or  more,  ivrite  the  figure  standing 
for  its  ones,  and  add  its  tens  with  the  units  of  the  next 
higher  order. 

Write  thewhole  sum  of  the  units  of  the  highest  order. 

PROOF.— Add  the  numhers  a  second  time,  in  a, 
different  order.  If  the  xvorh  is  correct,  the  result  will 
he  the  same  by  both  methods. 


rjtOJiljEMS. 

Add  and  pr- 

ove — 

(1.) 

(2.) 

(3.) 

(4.) 

(5.) 

4306 

172 

3334 

146 

4455 

4507 

903 

678 

504 

777 

8421 

129 

105 

810 

91 

6.  What  is  the  sum  of  563,  491  and  708  ?        Ans.  1762. 

7.  What  is  the  sum  of  423,  567  and  385? 

8.  Find  the  sum  of  785,  584,  175  and  145.      Ans.  1689. 


ADDITION. 


21 


9.  Find  the  sum  of  2385,  385  and  500.  Ans.  3270. 

10.  Find  the  sum  of  1603,  495,  1708  aud  793. 

11.  How  many  are  11063,  4461,  1030  and  309? 

12.  How  many  are  64004,  15300,  1008  and  488  ? 

13.  How  many  are  3560,  1246,  8556  and  2451  ? 

14.  How  many  are  2063  +  950  +  805  +  470  ?    Ans.  4288. 

15.  How  many  are  1623,  2045,  705  and  3435? 

16.  How  many  are  4000,  7891,  1631,  500  aud  19? 

17.  600  +  419  +  6663  +  1000  +  317  =  what  number? 

18.  1798  +  4444  +  6666  +  1333  +  122  =  how  many  ? 

19.  20120  +  3290  +  3167  +  532  +  499  =  how  many  ? 

20.  Add  138935,  113467  and  12506.  Ans.  264908. 

21.  Add  38394,  22957  and  601826.  Ans.  663177. 

22.  Add  2352  yards,  3800  yards  and  1785  yards. 

23.  How  many  dollars  are  13177  dollars,  7346  dollars  aud 
1275  dollars  ? 


24.  Upon  a  platform  car  there 
are  three  rough  blocks  of   stone. 
One  of  them  weighs  3470  pounds; 
another,  5362  pounds ;  and  the  third, 
2567  pounds.     What  is  the  weight  of  the  three  ? 

Ans.  11399  pounds. 

25.  Six  bales  of  cotton  contain,  respectively,  470,  480,  45d, 
500,  512  and  475  pounds.     What  is  their  entire  weight  ? 

26.  If  four  boxes  weigh,  respectively,  1618,  1450,  963  and 
1341  pounds,  what  is  their  entire  weight? 

27.  A  merchant  bought  at  one  time  513  barrels  of  flour ; 
at  another,  763  barrels ;  and  at  another,  1 347  barrels.  How 
many  barrels  did  he  buy  in  all  ?  Ans.  2623. 


2!J  ADDITION. 

28.  A  gardener  took  from  one  tree  348  pears  ;  from  another, 
316 ;  from  another,  159 ;  and  from  another,  96.  How  many 
did  he  take  from  the  whole  ?  Ans.  919. 

29.  In  a  certain  school  there  are  in  the  first  section  31 
pupils ;  in  the  second,  39  ;  in  the  third,  42  ;  in  the  fourth,  47  ; 
and  in  the  fifth,  64.  How  many  pupils  are  there  in  the  five 
sections?  Ans.  223. 

30.  Thompson  has  in  his  farm  150  acres ;  Reed  in  his 
farm,  317  acres ;  Howland  in  his  farm,  72  acres ;  and  two 
others  have  each  875  acres.     How  many  acres  have  they  all  ? 

31.  A  large  work  consists  of  five  volumes :  there  are  612 
pages  in  the  first  volume,  709  in  the  second,  691  in  the  third, 
and  1357  in  the  remaining  two  together.  How  many  pages 
in  the  entire  work  ?  Ans.  3369. 

32.  The  distance  from  New  York  to  Chicago  by  railroad  is 
911  miles ;  from  Chicago  to  Omaha,  401  miles ;  from  Omaha 
to  Ogden,  1101  miles;  from  Ogden  to  Sacramento,  743  miles; 
from  Sacramento  to  San  Francisco,  117  miles.  What  is  the 
whole  distance  from  New  York  to  San  Francisco  ? 

33.  A  lumber-dealer  has  37270  feet  of  boards  in  one  yard, 
9536  in  another,  45098  in  vessels,  and  8876  at  a  mill.  How 
many  feet  of  boards  has  he  in  all  these  places  ? 

34.  I  bought  a  house  for  6236  dollars ;  I  paid  869  dollars 
for  repairs,  and  300  dollars  for  painting  it.  For  what  price 
must  it  be  sold  to  gain  325  dollars  ?  Am.  7730  dollars. 

35.  The  sailing  distance  from  New  OHeans  to  Charleston  is 
1167  miles;  from  Charleston  to  Norfolk,  431  miles;  from 
Norfolk  to  New  York,  308  miles  ;  and  from  New  York  to  Bos- 
ton, 356  miles.  How  many  miles  must  a  steamer  sail,  that 
^hall  touch  at  all  these  places,  in  going  from  New  Orleans  to 

Boston?  Ans.  2262. 

36.  The  mariner's  compass  was  invented  in  China  1120 
years  before  Christ;  America  was  discovered  by  Columbus 
1492  years  after  Christ ;  and  steam  was  first  applied  by  Fulton 
to  ])ropclling  boats  315  years  after  the  discovery  of  America. 
How  many  years  after  the  invention  of  the  mariner's  compass 
was  steam  first  applied  to  propelling  boats  ? 


ADDITION.  23 

Copy,  add  and  prove — 

(37.)            (38.)             (39.)  (40.)  (41.) 

131         600         180  4811  11319 

256         176         316  1141  4121 

702           31          701  7891  3006 

91           45             9  123  723 

61           39         114  416  345 


42.  Alaska  contains  577,390  square  miles;  California,  188,981; 
and  Oregon,  95,274.  How  many  square  miles  do  the  three  con- 
tain? Ans.  861,645. 

43.  In  1868  there  were  employed  in  the  United  States,  as 
cultivators  of  the  land,  3,219,495  persons ;  as  day-laborers, 
960,000 ;  as  servants,  560,000 ;  as  mechanics  and  manufac- 
turers, 480,905  ;  and  as  merchants,  123,564.  How  many  per- 
sons were  employed  in  all  of  the  occupations  named? 

44.  The  area  of  America  is  15,977,000  square  miles;  of 
Europe,  3,954,000  square  miles;  of  Asia,  15,742,000;  of 
Africa,  11,734,000;  and  of  Oceanica,  3,700,000.  What  is  the 
area  of  the  whole  ?  Ans.  51,107,000  square  miles. 

45.  In  1870  the  population  of  America  was  85,950,000; 
of  Europe,  300,390,000;  of  Asia,  795,000,000;  of  Africa, 
174,240,000;  and  of  Oceanica,  30,102,000.  What  was  the 
population  of  the  whole  ? 


TEST    QUESTIONS. 

50.— 1.  What  is  Addition?  The  sum  or  amount?  A  sign?  The 
sign  of  addition  ?  The  sign  of  equality  ?  Illustrate  the  use  of  the  sign 
of  addition.     Of  the  sign  of  equality. 

2.  What  are  Principles  of  addition  ?  Why  cannot  5  dollars  and  4 
books  be  added  ?  Why  can  4  dollars  and  5  dollars  be  added  ?  Sho-w 
that  the  sum  of  numbers  is  the  same  in  whatever  order  they  are  added. 

3.  What  is  the  Rule  for  addition  ?  Why  are  the  numbers  in  addition 
•nritten  so  that  all  figures  of  the  same  order  shall  stand  in  the  same 
column?  Why  begin  at  the  right  to  add?  When  the  sum  of  any 
column  is  ten  or  more,  why  add  its  tens  with  units  of  the  next  higher 
order?     Give  the  proof  of  nddition.     The  renson  for  the  proof. 


24  SUBTRACTION. 

SECTION   IV. 

SUBTRACTION. 

51. — Ex.  1.  If  John  has  7  books,  liow  many  more  must  he 
obtain  to  have  12? 

2.  In  my  purse  there  "were  11  dollars ;  6  dollars  have  since 
been  taken  out.     How  many  remain  ? 

3.  William  is  17  years  old,  and  his  brother  is  8.     "NYhat  is 
the  difference  in  their  ages? 

4.  Subtract  by  2's  from  21  to  1. 

Solution.— 19,  17,  15,  13,  11,  9,  7,  5,  3,  1. 

5.  Subtract  by  3's  from  22  to  1.  By  3's  from  52  to  22. 

6.  Subtract  by  4's  from  32  to  4.  By  4's  from  60  to  32. 

7.  Subtract  by  5's  from  33  to  3.  By  5's  from  68  to  33. 

8.  Subtract  by  6's  from  34  to  4.  By  6's  from  70  to  34. 

9.  Subtract  by  7's  from  40  to  5.  By  7's  from  75  to  40. 

10.  Subtract  by  8's  from  46  to  6.     By  8's  from  94  to  46. 

11.  Subtract  by  9's  from  52  to  7.     By  9's  from  88  to  52. 

12.  What  is  the  unit  of  12  and  7?     What  is  the  unit  of  the 
difference  of  12  and  7  ? 

13.  What  is  the  unit  of  the  difference  of  13  dollars  and  9 
dollars  ? 

14.  Are  13  dollars  and  9  dollars  and  their  difference  similar 
or  dissimilar  numbers  ? 

15.  Why  can  you  not  subtract  9  books  from  13  dollars? 
Because  the  numbers  liave  no  common  unit. 

16.  Only  what  kind  of  a  number,  therefore,  can  be  taken 
from  another  number  ? 

17.  If  you  take  7  cents  from  16  cents,  what  luimber  will  be 
the  difference  ?    What  number  added  to  7  will  make  16  ? 

DEFINITIONS. 

52.  Subtraction  is  the  process  of  taking  one  number  from 
another. 

53.  The  Difference  is  the  result  obtained  by  the  subtraction. 


SUBTRACTION.  26 

54.  The  Minuend  is  the  number  from  which  the  subtraction 
is  to  be  made. 

55.  The  Subtrahend  is  the  number  which  is  to  be  subtracted. 

The  subtrahend  cannot  be  greater  than  the  minuend.  When  the  sub- 
trahend is  equal  to  the  minuend,  the  difference  is  0. 

5G.  The  Sig'u  of  Subtraction  is  — ,  and  is  called  minus. 
When  placed  between  two  numbers  it  denotes  that  the  one  on 
the  right  is  to  be  taken  from  that  on  the  left. 

Thus,  13  —  5  is  read,  thirteen  minus  five,  and  means  that  five  is  to  be 
taken  from  thirteen. 

Principles  of  Subtraction. 

57. — 1.   Only  similar  numbers  can  be  subtracted. 

2.  The  minuend,  subtrahend  and  difference  mxist  be  similar 
numbers. 

3.  The  sum  of  the  difference  and  subtrahend  must  equal  the 
minuend. 

C^SE   I. 

Wlieu  the  Units  of  the  Subtrahend  do  not  Exceed  those  of  the 
same  Orders  in  the  Minuend. 

58.— Ex.  1.  From  569  subtract  235. 

Minuend,       569  Solution. — For  convenience  in  subtracting, 

Subtrahend    235        write  the  figures  of  the  subtrahend  under  figures 

representing  the  same  orders  in  the  minuend. 
Difference,      004  Subtract   each    order   of  units  separately:  9 

ones  —  5  ones  are  4  ones,  which  write  for  the  ones  of  the  difference. 

6  tens  —  3  tens  are  3  tens,  which  write  for  the  ten^<  of  the  difference. 

5  hundreds  —  2  hundreds  are  3  hundreds,  which  writs  for  the  hun- 
dreds of  the  difference. 

Hence,  the  difference  is  3  hundreds  3  tens  4  units,  or  334. 


(2.) 

(3.) 

(4.) 

(5.) 

From       496 

873 

98 

3318  dollars, 

Subtract  365 

651 

54 

208  dollars. 

In  Ex.  5,  as  there  are  no  units  of  the  thousands'  order  in  the  subtra- 
hend, we  proceed  as  if  the  subtrahend  had  been  written  0208. 
.3 


26  SUBTRACTION. 

6.  A  person  born  in  the  year  1819  was  how  old  in  1871  ? 

7.  Johnson  had  765  dollars,  and  paid  away  351  dollars. 
How  many  dollars  had  he  left  ?  Ans.  414. 

8.  What  is  the  difference  between  3467  and  1352  ? 

9.  What  is  the  difference  between  25674  and  5473  ? 

10.  Andrew  Holt  bought  a  plantation  for  14630  dollars,  and 
sold  it  for  15750  dollars.  How  many  dollars  did  he  gain  by 
the  operation  ?  Ans.  1120. 

ca.se  II. 

T^licn  the  Units  of  one  or  more  Orders  of  the  Subtrahend  Ex- 
ceed those  of  the  Same  Orders  of  the  Minuend. 

59 —Ex.  1.  From  8662  take  4734. 

7.16.5.12.  Solution. — Write  the  numbers  as  in  the  pre- 

Mlnuend,        8G62       ceding  case. 

Subtrahend,   4734.  Since  4  ones  cannot  be  subtracted  from   2 

nQQQ       ones,  take  1  ten  from  the  6  tens,  leaving  5  tens; 

JJ  '  and  add  10  ones,  which  are  equal  to  the  1  ten, 

to  the  2  ones,  making  12  ones ;  12  ones  —  4  ones  are  8  ones,  which  write 
for  the  ones  of  the  difference. 

5  tens  —  3  tens  are  2  tens,  which  write  for  the  tens  of  the  difference. 

Since  7  hundreds  cannot  be  subtracted  from  6  hundreds,  take  1  thou- 
sand from  the  8  thousands,  leaving  7  thousands,  and  add  the  10  hundreds 
that  equal  the  1  thousand  to  the  6  hundreds,  making  16  hundreds;  16 
hundreds  —  7  hundreds  are  9  hundreds,  which  write  for  the  hundreds 
of  the  differeiice. 

7  thousands  —  4  thousands  are  3  thousands,  which  write  for  the  thou- 
sands of  the  difference. 

Hence,  the  diflcrence  is  3  thousands  9  hundreds  2  tens  8  units,  or  3928. 

The  explanation  may  be  abbreviated  thus :  2  +  10  are  12 ;  12  —  4  are 
8,  which  write.  6^1  are  5  ;  5  —  3  are  2,  which  write.  6  +  10  are  16 ; 
16  —  7  are  9,  which  write.     8  —  1  are  7  ;  7  —  4  are  3,  which  write. 

Ans.  3928. 

"We  begin  at  the  right  to  subtract,  so  tliat  when  any  figure  of  the  sub- 
trahend denotes  a  greater  number  tlian  tlie  corresponding  figure  of  the 
minuend,  1  may  be  taken  from  tlie  ne.\t  higher  order  of  the  minuend. 

Subtrahend,    4734  Proof.— To  test  tlie  solution,  we  add  the 

Difference       39'^ 8  difference  and  subtrahend,  and  since  the  result 

-  —  is  equal  to  the  minuend,  the  work  is  known  to 

Minu&ixd,         8662  be  correct.  (Art.  57-3.) 


SUBTRACTIOX.  27 

Solve  and  prove — 

(2.)                   (3.)                     (4.)  (5.) 

From      4107            6483  1922  5r6'5' pounds, 

8vhiTad  3058              938  1415  6718  pounds. 

6.  What  is  the  difference  between  1634  and  1329  ? 

7.  I  bought  goods  for  9363  dollars,  and  sold  them  for  10382 
dollars.     How  much  did  I  gain  by  the  transaction  ? 

Ans.  1019  dollars. 

60.  Rule  for  Subtraction.— Tr77Y<^  the  subtrahend  under 
the  minuend,  to  that  figures  of  the  same  order  shall 
stand  in  the  same  column. 

Begin  at  the  right,  subtract  tJi  e  units  of  each  order 
of  the  subtrahend  from  units  of  the  same  order  of  the 
minuend,  if  possible,  and  write  thejiifference  beneath. 

When  the  units  of  any  order  of  the  minuend  are 
less  than  those  of  the  same  order  of  the  subtrahend, 
increase  their  7iumber  by  adding  ten,  the  value  of  a 
unit  tahen  from  the  next  higher  order  of  the  minu- 
end, and  subtract;  then  consider  the  units  of  that 
higher  order  of  the  minuend  one  less. 

PROOF. — Add  the  differeitce  and  subtrahend.  If 
the  ivorh  is  connect,  the  result  will  equal  the  lyvhvuend. 

PROBIjEMS. 

Solve  and  prove — 

(1.)                    (2.)  (3.)  (4.) 

JVom  4592            9863  34061  46603  soldiers, 

Take   3683              416  12341  928  soldiers. 

5.  How  many  are  69351  less  40402?  Ans.  28949. 

6.  How  many  are  32443  less  965  ? 

7.  How  many  are  17630  less  8542?  Ans.  9088. 

8.  What  is  the  difference  between  92334  and  13403? 

9.  What  is  the  difference  between  46304  and  37413? 

10.  How  much  more  is  32045  than  862?         Ans.  31183. 

11.  How  much  less  is  40920  th^u  91605?        Ans.  50685, 


28  SUBTRACTION. 

12.  36841  —  27777  =  what  number?  Ans.  9064. 

13.  81710  —  70809  =  what  number  ?  Ans.  10901. 

14.  What  is  the  difference  between  4000  and  1321  ? 

3.9.9.10.  Solution. — As  there  are  no  units  expres.=ed 

Minuend,       4^000       in  the  three  lower  orders  of  the  minuend,  re- 

Subtrahend    132 1        duce  one  of  the  four  thousands  to  hundreds, 

then  one  of  the  hundreds  to  tens,  and  then  one 

Difference,     2b  7  J       ^^  ^j^g  ^^^^  ^^  ^^^^_    r^^^  minuend  will  then  be 

3  thousands  9  hundreds  9  tens  10  ones,  from  which  the  subtrahend  can 
easily  be  taken. 

Then,  10  ones  —  1  one  are  9  ones,  which  write  for  the  ones  of  the 
difference.  9  tens  —  2  tens  are  7  tens,  which  write  for  the  tens  of  the 
difference.  9  hundreds  —  3  hundreds  are  6  hundreds,  which  write  for 
the  hundreds  of  the  difference.  3  thousands  —  1  thousand  are  2  thou- 
sands, which  write  for  the  thousands  of  the  difference.  Hence,  the 
answer  is  2679. 

Solve  and  prove —  . 

(15.)           (16.)  (17.)-  (18.) 

From  34-00  m^n,          5002  8003  20000  hn^Yida, 

Take  2306    "     2151  7004  4242      " 

19.  Subtract  199  from  1000.  Ans.  801. 

20.  Subtract  3003  from  30030.  An&.  Ti^ll. 

21.  How  many  are  19000  men  less  567  men? 

22.  How  many  yards  are  31005  yards  less  1777  yards? 

23.  How  many  tons  are  51067  tons  less  45075  tons  ? 

24.  From  J  00000  take  16898.  Ans.  83102. 

Find  the  difference — 

25.  Between  36512  and  18735.  Ans.  Villi. 

26.  Between  12701  and  4110. 

27.  Between  7125  and  3647.  Ans.  3478. 

28.  Between  38217  and  9548.  Ans.  28669. 

29.  Between  1000000  and  107701. 

30.  Between  500807  and  480705.  Am.  20102. 

31.  If  a  man  should  buy  a  farm  for  8000  dollars,  and  should 
make  a  payment  of  5925  dollars  upon  it,  how  many  dollars 
would  he  still  owe?  Ans.  2075. 

32.  The  Pilgrims  landed  at  Plymouth  in  the  year  1620. 


SUBTRACTION.  29 

How  long  was  it  from  that  time  to  the  Declaration  of  Indepen- 
dence, in  the  year  1776? 

33.  A  merchant  bought  63100  bushels  of  corn,  and  sold 
17734  bushels.     How  many  bushels  had  he  left  ? 

34.  I  bought  a  piece  of  property  for  95565  dollars,  and  sold 
it  for  100000  dollars.   What  was  the  gain  ?     Ans.  4435  dollars. 

35.  In  a  certain  election  A  received  25316  votes  and  B 
21925  votes.     What  was  A's  majority  ?  ^ns.  3391. 

36.  By  selling  a  farm  for  7535  dollars  I  gained  1675  dollars. 
What  v.-as  its  cost  ? 

37.  At  a  certain  election  the  successful  candidate  received 
25316  votes,  which  was  3191  votes  more  than  were  given  to  the 
rival  candidate.     How  many  votes  did  the  latter  receive  ? 

38.  The  population  of  a  town  is  2763 ;  ten  years  ago  it  was 
2595.     What  was  the  increase  in  the  ten  years  ?     Ans.  168. 

39.  A  mill  sawed  427563  feet  of  boards  one  year  and  385895 
the  year  before.  How  many  more  feet  did  it  saw  in  the  one 
year  than  in  the  other?  Ans.  41668. 

40.  St.  Peter's  Church,  Rome,  has  standing-room  for  54000 
persons,  and  the  New  York  Academy  for  1326  persons.  How 
many  more  persons  can  stand  at  the  same  time  in  the  one  than 
in  the  other  ? 

41.  There  are  41823360  seeds  in  a  bushel  of  timotliy,  and 
16400960  seeds  in  a  bushel  of  clover.  How  many  more  seeds 
in  a  bushel  of  the  one  than  in  a  bushel  of  the  other  ? 


TEST    QUESTIONS. 
61. — 1.  What  is  Subtraction?     The  difference?     The   minuend? 
The  subtrahend?     When  the  subtrahend  and  minuend  are  equal,  what  is 
their  diflerence  ?     What  is  the  sign  of  subtraction  ?     Ilhistrate  its  use. 

2.  What  are  the  Principles  of  subtraction?  Why  cannot  5  apples 
be  subtracted  from  8  dollars  ?  If  the  minuend  and  subtrahend  express 
dollars,  what  will  the  difference  express?  If  the  minuend  is  8,  what 
must  be  the  sum  of  the  subtrahend  and  difference? 

3.  What  is  the  Rule  for  subtraction  ?  Why  is  the  subtrahend  written 
under  the  minuend  so  that  figures  shall  stand  in  the  same  column  ? 
Why  begin  at  the  right  to  subtract?  .What  is  the  method  of  proof? 
When  the  subtrahend  and  difference  are  given,  how  may  the  minuend 
be  found  ? 


30  BE  VIEW   PROBLEMS. 

SECTION    V. 
■      REVIEW  PROBLEMS. 

MENTAL,    EXEJtCISES. 

62, — Ex.  1.  How  many  tens  and  ones  in  96?     In  78? 

2.  How  many  hundreds,  tens  and  ones  in  365?  In  431? 
In  987? 

3.  How  many  thousands  and  units  in  55625?     In  107308? 

4.  John  has  14  dollars,  William  10  dollars  and  Henry  9 
dollars.     How  many  dollars  have  they  all  ? 

5.  A  farmer  has  26  cows  in  one  field,  15  in  another  and  23 
in  a  third.     How  many  cows  has  he  in  the  three  fields  ? 

Solution. — He  has  as  many  cows  in  the  three  fields  as  the  sum  of  26, 
15  and  23.  26  +  10  =  36  ;  36  +  5  =  41 ;  41  +  20  =  61 ;  61  +  3  =  64. 
Therefore  he  has  64  cows  in  the  three  fields. 

6.  A  man  paid  18  dollars  fiar  a  coat,  33  dollars  for  an  over- 
sack  and  9  dollars  for  pantaloons.  How  much  did  he  pay  for 
all? 

7.  A  drayman  moved  41  boxes  on  one  day,  17  on  another 
day,  34  on  a  third  day  and  23  on  a  fourth  day.  How  many 
boxes  did  he  move  in  all? 

8.  A  certain  orchard  contains  28  apple  trees,  18  pear  trees, 
32  peach  trees  and  1 1  cherry  trees.  How  many  trees  does  it 
contain  altogether? 

9.  If  I  had  9  dollars  more  I  should  have  27  dollars.  How 
much  money  have  I? 

10.  John  had  48  cents,  but  has  since  spent  29  cents.  How 
many  cents  has  he  remaining  ? 

Solution.— He  has  as  many  cents  remaining  as  the  difference  between 
48  and  29.  48  —  9  =  39  ;  39  —  20  =  19.  Therefore  he  has  19  cents 
remaining. 

11.  Robert  is  15  years  old,  and  his  father  is  53  years  old. 
What  is  the  difiercnce  in  their  ages  ? 

12.  Susan  is  13  years  old.  In  how  many  years  will  she  be 
42  years  old? 

13.  AVillio  gathered  27  nuts  in  the  forenoon,  and  in  the  after- 


REVIEW    PROBLEMS.  31 

noon  as  many  more  less  19.     How  many  did  he  gather  in  the 
afternoon  ? 

14.  A  drover  bought  16  sheep  at  one  time  and  112  at 
another ;  he  then  sold  18.     How  many  had  he  left  ? 

15.  I  had  54  cents  in  my  purse,  but  took  out  15  cents  at  one 
time  and  16  cents  at  another.  How  many  cents  then  remained 
in  the  purse  ? 

WRITTEN   EXERCISES. 

63. — Ex.  1.  A  man  paid  700  dollars  for  a  span  of  horses, 
450  dollars  for  a  carriage  and  175  dollars  for  a  double  harness. 
What  sum  did  he  pay  for  the  whole?         Ans.  1325  dollars. 

2.  What  is  the  sum  of  one  million  five  hundred  seventy-five 
thousand  three  hundred  twenty-two,  plus  four  hundred  nineteen 
thousand  three  hundred  sixty-five,  plus  one  hundred  thirty-two 
thousand  three  hundred  fifty-five  ? 

3.  I  bought  some  property  for  5390  dollars,  and  sold  it  for 
6585  dollars.     How  much  did  I  gain  ?       Ans.  1195  dollars. 

4.  If  the  smaller  of  two  numbers  is  5390  and  their  diflference 
1195,  what  is  the  larger  number? 

5.  Mount  St.  Elias  is  17900  feet  in  height  and  Mount  Wash- 
ington 6284  feet.  How  much  higher  is  Mount  St.  Elias  than 
Mount  Washington?  Ans.   11616  feet. 

6.  If  the  minuend  is  four  hundred  seventeen  thousand  seven 
hundred  twelve,  and  the  subtrahend  is  one  hundred  thousand 
ninety-three,  what  is  the  difference  ?  Ans.  317619. 

7.  A  man's  salary  is  1200  dollars,  and  his  income  from  other 
sources  is  655  dollars.  His  expenses  are  850  dollars.  How 
much  is  his  net  income  ? 

Solution.  —  His  income  must  be 

1200  dollars.        the  sum  of  his  salary  and  his  income 

655        "  from  other  sources.      1200  +  65-5  = 

To^r  J   n  ^^'^^  dollars. 

Income,  looo  dollars.  tj-       ^.  •  ^  u   i  •    • 

'  His  net  income  must  be  his  income 

Kvjoenses,  850        "  less  his  expenses.      1855    dollars  — 

Net  Income,   1005  dollars.       850  dollars  =  1005  dollars.     His  net 

income,  therefore,  is  1005  dollars. 


32  REVIEW  PROBLEMS. 

8.  A  merchant  bought  goods  to  the  amount  of  7835  dollars. 
He  then  sold  a  part  for  9563  dollars,  and  the  balance  for  361 
dollars.     How  much  did  he  gain  by  the  transaction  ? 

9.  The  great  bell  of  Moscow  weighs  432000  pounds,  and  the 
bell  of  City  Hall,  New  York,  weighs  22300  pounds.  What  is 
the  difference  in  their  weight  ? 

10.  A  grain-dealer  had  9867  bushels  of  corn.  He  sold 
3479  bushels  at  one  time  and  5372  bushels  at  another  time. 
How  many  bushels  had  he  then  left  ? 

Solution.— If  he  sold  3479 

34-79  bushels.       9867  bushels.        bushels  at   one  time  and  5372 

5372        "  8851         "  bushels    at    another    time,    he 

oozr  -f  ^      1    1  -I  r\-i  /:>  i       i    i  sold  in  all  3479  bushels  -|-  5372 

8851  bushels.      10 lb  bushels.       ,     ,   ,        00-11,1.1 

bushels,  or  880I  bushels. 

If  he  had  9867  bushels,  and  sold  8851  bushels,  he  had  left  9867  bush- 
els —  8851  bushels,  or  1016  bushels.  Ans.  1016  bushels. 

11.  Kaspar  sold  516  barrels  of  flour  at  one  time,  419  bar^jels 
at  another  time,  and  had  316  remaining.  How  many  barrels 
had  he  at  first  ?  Ans.  1251. 

12.  A  man  had  16745  acres  of  land.  To  A  he  sold  5304 
acres  and  to  B  6319  acres.     How  much  had  he  then  unsold? 

13.  Andrew  Jones  bought  a  piece  of  land  for  18068  dollars. 
He  sold  a  part  for  9563  dollars,  and  the  remainder  for  9385 
dollars.     How  much  did  he  gain  by  the  operation  ? 

14.  A  man  had  15675  dollars ;  afterward  he  expended  5000 
for  a  house  and  1225  for  furniture,  and  then  earned  963  dollars. 
How  much  money  had  he  then  ?  Aiis.  10413  dollars. 

15.  The  sailing  distance  from  New  York  to  Cape  Horn  is 
7232  miles ;  from  Cape  Horn  to  Canton,  8840  miles ;  from 
Canton  to  Cape  Good  Hope,  7000  miles  ;  and  from  Cape  Good 
Hope  to  New  York,  6790  miles.  The  sailing  distance  from 
Canton  to  San  Francisco  is  6090  miles,  and  the  distance  by 
railroad  from  San  Francisco  to  New  York,  3273  miles.  How 
much  shorter  is  the  route  from  Canton  to  New  York,  by  way 
of  San  Francisco,  than  by  cither  Cape  Good  Hope  or  Cape 
Horn?         Aixs.  4427  miles  shorter  than  by  Ca]>e  Good  Hope; 

6709  miles  shorter  tlian  by  Cape  Horn. 


MULTIPLICATION.  83 

SECTION    VI. 

MUL  TIPLICA  TlOJf. 

64. — Ex.  1.  If  a  boy  take  from  a  library  4  books  each  day 
for  3  days,  how  many  times  does  he  take  4  books  ?  How  many 
books  does  he  take  in  all  ? 

2.  Four  boys  have  each  6  cents.  How  many  cents  have 
they  in  all  ?  How  many  times  must  6  cents  be  taken  to  make 
the  number  they  have  in  all  ? 

3.  What  is  the  difference  between  6  cents  -f-  6  cents  +  6 
cents  +  6  cents  and  4  times  6  cents  ? 

4.  How  many  times  9  cents  will  3  pencils  cost  at  9  cents 
each? 

5.  Give  tilt  2's  from  one  2  to  six  2's. 

Solution. — One  2  is  2,  two  2's  are  4,  three  2's  -are  6,  four  2's  are  8, 
five  2's  are  10,  and  six  2's  are  12. 

6.  Give  the  2's  from  seven  2's  to  twelve  2's. 

7.  Give  the  3's  from  three  3's  to  twelve  3's. 

8.  Give  the  4's  from  five  4's  to  twelve  4's. 

9.  Give  the  5's  from  five  5's  to  twelve  5's. 

10.  Give  the  6's  from  six  6's  to  twelve  6's. 

11.  Give  the  7's  from  seven  7's  to  twelve  7's. 

12.  Give  the  8's  from  eight  8's  to  twelve  8's. 

13.  Give  the  9's  from  nine  9's  to  twelve  9's. 

14.  Give  the  lO's  from  ten  lO's  to  twelve  lO's. 

15.  Give  the  ll's  from  eleven  ll's  to  twelve  ll's. 

16.  How  many  are  5  times  7?  4  times  8  dollars?  Is  the 
number  which  denotes  the  times  another  number  is  taken  ab- 
stract or  concrete  ? 

17.  When  7  dollars  are  taken  6  times,  what  is  the  resuh  ? 
To  which  of  the  given  numbers  is  the  result  similar  ? 

18.  Is  the  result  the  same  when  we  take  8  five  times  as  when 
we  take  5  eight  times  ? 

19.  How  many  are  10  times  6?     10  times  9? 

20.  How  many  are  11  times  5?     11  times  10? 

21.  How  many  are  12  times  12?     12  times  11  ? 


34  MULTIPLICATION. 

DEFINITIONS. 

65.  Multiplication  is  the  process  of  taking  one  of  two  num- 
bers as  many  times  as  there  are  units  in  the  other. 

66.  The  Multiplicand  is  the  number  taken  or  multiplied. 

67.  The  Multiplier  is  the  number  which  shows  how  many- 
times  the  multiplicand  is  to  be  taken. 

68.  The  Product  is  the  result  of  the  multiplication. 

69.  The  multiplier  and  multiplicand  are  called  Factors  of 
the  product. 

When  the  factors  are  more  than  two,  the  multiplication  is  called  Con- 
tinued Multiplication,  and  the  result  a  Continued  Product. 

70.  The  Si^  of  Multiplication  is  X ,  and  is  read,  times,  or 
multiplied  by.  When  placed  between  two  numbCTs,  it  denotes 
that  they  are  to  be  multiplied  together. 

Thus,  8  X  5  is  read,  e'ght  multiplifd  hy  five,  ov five  times  eight. 

Principles  of  Multiplication. 

71. — 1.   Tlie  multiplier  is  always  a7i  abstract  number. 

2.  The  product  and  the  multiplicand  are  similar  numbers. 

3.  In* finding  the  product  of  tivo  factors,  either,  abstractly  con- 
sidered, may  be  used  as  the  multiplier. 

ca.se  I. 

Wlien  the  Multiplier  consists  of  but  One  Order  of  Units. 

72.— Ex.  1.  Multiply  3427  by  6. 

Multiplicand,      34^^7  Solution. — For   convcnicnoo  write   the 

Multiplier  6       multiplier  under  the  multiplicand,  and  begin 

Product,    '        20562       -tf;-gl!t  to  multiply 

'  6  times  7  ones  are  42  ones,  or  4  tens  and 

2  ones.     Write  the  2  ones  for  the  ones  of  the  product,  and  reserve  tlie 
4  tens  to  add  to  the  product  of  the  tens. 

()  times  2  tens  are  12  tens,  and  12  tens  +  4  tens  are  16  tens,  or  1  hun- 
dred, and  6  tens.  Write  the  6  tens  for  the  tens  of  the  product,  and 
re.'^e^vl■  tin-  1  luuidred  to  add  to  llic  product  of  tiie  hundreds. 


M ITL  TIPLICA  TION.  35 

6  times  4  hundreds  are  24  hundreds,  and  24  hundreds  +  1  liundred  are 
25  hundreds,  or  2  thousands  and  5  hundreds.  Write  the  5  hundreds 
for  the  hundreds  of  the  product,  and  reserve  the  2  thousands  to  add  to 
the  product  of  the  thousands. 

6  times  3  thousands  are  18  thousands,  and  18  thousands  +  2  thousands 
are  20  thousands,  or  2  ten-thousands  and  0  thousands,  which  write. 

Hence,  the  product  is  2  ten-thousands  0  thousands  5  hundreds  6  tens 
2  ones,  or  20562. 

We  begin  at  the  right  to  multiply,  so  that  when  the  product  exceeds 
nine,  we  may  add  its  tens  to  the  next  product. 

The  explanation  may  be  abbreviated  thus :  6  times  7  are  42 ;  write 
the  2  and  reserve  the  4.  6  times  2  are  12,  and  4  are  16  ;  write  the  6  and 
reserve  the  1.  6  times  4  are  24,  and  1  is  25 ;  write  the  5  and  reserve  the 
2.     6  times  3  are  18,  and  2  are  20,  which  write.  Ans.  20562. 

The  correctness  of  the  work  may  be  tested  or  proved  by  carefully  re- 
viewing the  whole  process. 

Solve  and  prove — 

(2.)                  (3.)  (4.)  (5.) 

MuUiphj  645            916  men.  1125  3246  yards. 

By          _J^          5                       _J  _3 

6.  Multiply  30874  by  6.  Ans.  185244. 

7.  What  is  the  product  of  73121  by  7?         Ans.  511847. 

8.  What  will  8  flirms  cost  at  13035  dollars  each? 

9.  How  many  square  miles  in  9  townships,  if  each  -contain 
24375  square  miles  ?  ^ns.  219375. 

CASE   II. 
When  the  Multiplier  consists  of  More  than  One  Order  of  Units. 

73— Ex.  1.  Multiply  962  by  34. 

Multiplicand        962  Solution.— Since  34  is  equal  to  3  tens  +  4 

"'              '  ones,  34  times  962  must  equal  4  times  962  -\- 

Multiplier,        J^       3  t^„g  times  962. 

Partial  j      3848  4  times  962,  or  the  product  of  962  by  the 

Products  1  2886  '^"^^  °^  t^^^  multiplier,  is  3848,  the  first  par- 

7                  Qc)iynQ  ^^^^  product. 

1  rodtict,            J^/U^  3  tgj^g  ^  2  ones  are  6  tens,  which  write  for 

the  tens  of  the  second  partial  product.  3  tens  X  6  tens  are  18  tens  of 
tens,  or  18  hundreds,  or  1  thousand  and  8  hundreds;  write  the  8  hun- 
dreds for  the  hundreds  of  the  partial  product,  and  reserve  the  1  thou- 
sand for  the  thousands.     3  tens  X  9  hundreds  are  27  tens  of  hundreds, 


36  MUL  TIPLICA  TION. 

or  27  thousands,  and  27  thousands  -|-  1  thousand  are  28  thousands,  or  2 
ten-thousands  and  8  thousands,  which  write  in  the  partial  product, 
making  the  second  partial  product  2886  tens,  or  28860. 

The  entire  product,  or  the  sum  of  the  partial  products,  3848  -\-  28860, 
is  32708. 

The  explanation  may  be  abbreviated,  thus :  962  X  4  is  3848  units.  3 
times  2  are  6,  which  write  for  the  tens.  3  times  6  are  18 ;  write  the  8 
for  the  hundreds,  and  reserve  the  1  for  the  thousands.  3  times  9  are  27, 
and  1  are  28,  which  write,  making  2886  tens. 

Adding  the  partial  products,  we  have  32708,  the  answer  required. 

Proof. — Prove  the  solution  by  reversing  the  order  of  the 
factors  in  the  multiplication,  taking  the  34  for  the  multipli- 
cand and  the  962  for  the  multiplier. 

The  result  being  the  same  as  at  first  obtained,  the  work  is 
presumed  to  be  correct,  since  the  product  of  two  factors  is 
the  same,  whichever  is  taken  as  the  multiplier.  (Art.  71 — 3.) 


32708 

Solve  and  prove — 

(2.)  (3.)  (4.)  (5.) 

Multiply  8125  813  3104  614 

By  _^  j55  16  207 

4298 

In  Ex.  5  we  omit  to  multiply  by  the  0  tens,  since  1228 

the  product  of  any  number  multiplied  by  0  is  0. 

6.  What  is  the  product  of  763  by  305?         Am.  232715. 

7.  What  is  the  product  of  706  by  408  ? 

8.  What  is  the  product  of  403  by  62  ?  Ans.  24986. 

C^SE   III, 
When  either  Factor  has  One  or  More  Ciphers  on  the  Right. 

74.— Ex.  1.-  Multiply  465  by  100. 
Multiplicand,  465  Solution. — Since   10  units  of  any  order 

Multiplier,  100     ^^^  always  equal   to  1   of  the  next  higher 

r,     J     ,  //^r:nr}      ^^'^^^  (Art.  34—1),  the  writing  of  a  cipher  on 

Jrodud,  40000      ^j^^  ^j^j^j   ^^  ^  number,  whk'h  removes  its 

figures  each  an  order  to  the  left,  must  multiply  it  by  10.  In  like  manner, 
the  writing  of  two  ciphers  on  the  right  of  a  number  nnist  multiply  it  by 
100;  the  writing  of  three  ciphers  must  multiply  it  by  1000,  etc. 


MULTIPLICATION.  37 

Hence,  to  465  X  1>  which  is  465,  we  annex  two  ciphers,  and  have  100 
times  the  number,  or  46500,  the  answer  requii'ed. 

2.  Multiply  465  by  500. 

Solution.— Since  500  is  100  times  5,  500 
Multiplicand,      465  times  465  must  be  100  times  5  times  465. 

Multiplier,  500  5  times  465  is  2325,  and  100  times  as 

Product,  232500      """"^^  ^'  ^^f  """^  ^'^"^  "?^'^'  ''""'^''^' 

or  232500,  the  answer  required. 

3.  Multiply  4650  by  500. 

Solution. — 5  times  4650  is  5  times 
Multiplicand,     4650  455  with   a  cipher   annexed,  or  23250; 

MuUijilier,  500         and  100  times  as  much  is  23250  with  two 

Product,  2325000      "P^^!"'  annexed,  or  2325000,  the  answer 

required. 


Multiply — 

4.  516  by  10.    Am.  5160. 

5.  1302  by  100. 

6.  95  by  1000. 

7.  254  by  600. 

8.  75  by  300.   Am.  22500. 


How  many  are — 
9.  40  times  560?       Am.  22400. 

10.  80  times  3400? 

11.  200  times  500?    ^ns.  100000. 

12.  120  times  4110? 

13.  1000  times  1000? 


14.  If  you  can  travel  12  miles  in  one  hour,  how  far,  at  the 
same  rate,  can  you  travel  in  100  hours? 

15.  What  will  45  casks  of  molasses  cost  at  5*0  dollars  each  ? 

16.  If  there  are  640  acres  in  a  square  mile,  how  many  acres 
are  there  in  150  square  miles  ?  Ans.  96000. 

75.  Rule  for  Multiplication.— 7/  the  -multiplier  consists  of 
one  order  of  units,  multiply  each  order  of  the  multi- 
plicand, beginning  at  the  right,  by  the  multiplier. 
Write  the  units  of  each  result  in  the  product,  and  re- 
serve the  tens,  if  any,  to  be  added  to  the  next  result. 

If  the  Tnultiplier  consists  of  more  than  one  order  of 
units,  multiply  each  order  of  the  inultiplicand  by 
each  order  of  the  multiplier,  ivrite  the  right-hand 
figure  of  each  partial  product  under  the  order  of 
the  multiplier  used,  and  add  the  partial  products. 


38 


MUL  TIP  Lie  A  TION. 


If  either  factor  has  one  or  more  ciphers  on  the 
right,  multiply  without  regard  to  these  ciphers,  and 
annex  to  the  result  as  inany  ciphers  as  are  on  the 
right  of  both  the  factors. 

PROOF.— Revieiv  the  ivorh,  or  reverse  the  order  of 
the  factors  and  inultiply.  If  the  work  is  correct,  the 
result  will  he  the  same  hy  both  methods. 


PROBLEMS. 


Multiply  and  prove — 

1.  216  by  8.  Arts.  1728. 

2.  405  by  9. 

3.  1315  by  6.  Am.  7890. 

4.  116  by  1000. 

5.  413  by  70.  Ans.  28910. 

6.  555  by  4. 

7.  4163  by  7.  Am.  29141. 

8.  3162  by  11. 

9.  51003  by  90. 

10.  18300  by  18. 

11.  738  by  235. 


12.  756  by  72. 

13.  3216  by  5. 

14.  248  by  19. 

15.  160  by  30. 

16.  365  by  37. 

17.  1040  by  11. 

18.  4561  by  603. 

19.  11140  by  13. 

20.  5704  by  974. 

21.  4402  by  222. 

22.  4561  by  4005. 


Am.   54432. 


Am.   4712. 


Am.   13505. 


Am.  2750283. 


Am.  5555696. 


23.  How  many  are  312  times  4144? 

24.  HoAV  many  are  999  times  345  ? 


Am.   1292928. 


345000  =  345  X  1000 
345  =  345  X    1 
999 


Solution. — Since  1000  times  any 
number,  less  once  the  number,  must 
be  999  times  tlie  number,  we  here 
abridge  the   solution    by   taking    the 


344655  =  345  X 

rnnhi()licaiul  1000  times,  or  once  too  many,  by  annexing  three  ciphers, 
and  tbL'ii  subtracting  the  multiplier.  This  method  of  abridgment  applies 
whenever  the  nniltiijlier  is  1  less  than  100,  1000,  10000,  etc. 


25.  4573  X  99  =  what  number?  Am.  AbT121. 

26.  13i6  X  999  =  what  number? 

27.  1230  X  9999  =  what  number?  Am.  12298770, 

28.  What  is  the  product  of  1036  by  990? 


MVL  TIPLICA  TION.  cM 

29.  What  is  the  product  of  4455  by  105  ? 

445500  =  4455  X  100  Solution.— Here  the  solution  may 

Qc^a>>^  r //^^v         ^       ^'^  abridged  by  writing  the  product  of  " 

'^^ 4455  by  the  1  hundred   at  once,  by 

4(>777o  =4455  X  105  annexing  two  ciphers,  and  writing 
under  it  the  product  of  4455  by  the  5  ones,  and  then  adding  the  two 
partial  products. 

30.  What  is  the  product  of  6307  by  1003  ? 

31.  If  5362  feet  of  boards  can  be  sawed  in  a  mill  in  one 
day,  how  many  feet  can  be  sawed  in  it  in  313  days  ? 

Ans.  1678306. 

32.  At  125  dollars  a  month,  how  many  dollars  can  be  earned 
in  12  months?  Ans.  1500. 

33.  What  will  3158  tons  of  coal  cost  at  8  dollars  per  ton? 

Q  7  r  o  Solution. — At  8  dollars  a  ton,  3158  tons  of  coal  will  cost 

o        3158  times  8  dollars,  which  is  equal  to  8  times  3158  dollars, 


or  25264  dollars. 


25264  8  dollars  is  the  true  multiplicand,  but  since  8  times  3158 

gives  the  same  product  as  3158  times  8,  we,  for  convenience,  in  the 
solution  consider  both  factors  as  abstract  numbers,  and  make  the  smaller 
factor  the  nuiltiplier. 

34.  What  will  8344  yards  of  cloth  cost  at  6  dollars  a  yard  ? 

Ans.  50064  dollars. 

35.  How  many  oranges  in  47  boxes  when  each  box  contains 
279  oranges?  ^ns.  13113. 

36.  Two  factors  are  7312  and  7000.     What  is  their  product? 

Ans.  51184000. 

37.  There  were  6  drawers  in  a  desk,  8  compartments  in  each 
drawer,  and  87  dollars  in  each  compartment.  How  many 
dollars  did  the  desk  contain  ? 

Solution. — Since  in   the    desk 
87  dollars.  there  were  6  drawers,  and  each  had 

48  No.  of  compartments.        «  compartments,  there  were  in  the 

desk  6  times  8,  or  48  compartments. 

^"^^  Since    there  were   48    compart- 

348  '  ments  in  the  desk,  and  87  dollars 

417 6  dollars  ^"   each,   the    desk    contained    48 

times  87  dollars,  gr  4176  dollars. 


40  MULTIPLICATION. 

38.  What  is  the  continued  product  of  the  factors  6,  8  and  87? 

39.  If  15  pounds  of  hay  are  required  by  1  horse  for  1  day, 
.  how  many  pounds  are  required  by  5  horses  for  30  days  ? 

40.  How  many  gallons  in  1025  casks,  if  each  cask  contain 
63  gallons  ?  Ans.  64575, 

41.  In  a  bushel  of  rye  are  888390  seeds.  How  many  seeds 
are  there  in  25  bushels  ?  ■    Am.  22209750. 

42.  If  the  w^eight  of  a  cubic  foot  of  white-oak  wood  is  43 
pounds,  what  is  the  weight  of  128  cubic  feet? 

43.  What  is  the  continued  product  of  the  factors  12,  420 
and  310?  Am.  1562400. 

44.  How  many  yards  of  cloth  in  32  bales,  each  bale  having 
121  pieces,  and  each  piece  31  yards?  Am.  120032. 

45.  What  is  the  population  of  a  State  containing  8320  square 
miles,  if  each  square  mile  has  91  inhabitants  ? 

46.  If  Pennsylvania  -'ontain  46000  square  miles,  what  will 
be  its  population  at  75  persons  to  a  square  mile  ? 

Ans.  3450000  persons. 

47.  Sound  moves  1142  feet  in  a  second.  How  far  will  it 
move  in  one  hour,  or  3600  seconds?  Am.  4111200  feet. 

48.  If  a  railroad,  1035  miles  in  length,  should  obtain 
government  aid  to  the  amount  of  52400  dollars  per  mile,  what 
would  be  the  entire  amount  received  ? 


TEST    QUESTIONS. 
<6. — 1.  What  is  MtTLTiPLicATioN ?    The  product?     The  factors  of  a 
product?     The   multiplicand?     The  multiplier?     What  does  the  sign 
of  multiplication  denote  when  written  between  two  numbers? 

2.  What  are  the  Principles  of  multiplication  ?  Show  that  the  mul- 
tiplicand and  product  are  similar  numbers.  Why  must  the  multiplier 
be  regarded  always  as  an  abstract  number  ?  Show  that  the  product  is 
the  same  in  whatever  order  its  factors  are  used. 

3.  Recite  the  Rule  for  multiplication.  Give  the  proof.  Why,  ui 
multiplication,  do  you  begin  at  the  right  to  multiply?  When  the  nnil- 
tiplier  consists  of  more  than  one  order  of  units,  how  many  partial  prod- 
ucts may  there  be  ?  What  will  be  the  unit  of  the  partial  product  when 
the  multiplier  is  a  number  of  simple  units?  When  the  multiplier  is  a 
number  of  tens  ?  When  there  are  partial  products,  how  do  you  obtain 
the  entire  product? 


DIVISION. 


41 


SECTION    VII. 


Dirisio.Y. 

77. — Ex.  1.  How  many  barrels,  holding  3  bushels  each,  wiil 
be  required  to  hold  30  bushels  of  apples  ? 

2.  If  a  farmer  should  raise  21  bushels  of  potatoes,  and 
should  wish  to  put  them  into  barrels  holding  3  bushels  each, 
how  many  barrels  would  be  required  ? 

3.  If  you  have  28  apples,  and  wish  to  distribute  them 
equally  among  7  boys,  how  many  can  you  give  to  each  boy  ? 

4.  How  many  times  are  9  bushels  contained  in  36  bushels  ? 
What  is  the  product  of  9  bushels  by  4  ?    . 

5.  How  can  you  show  that  9  bushels  are  contained  in  36 
bushels  4  times  ? 

6.  When  you  distribute  35  apples  equally  among  7  boys,  do 
you  find  how  many  times  7  boys  are  contained  in  35  apples, 
or  do  you  find  one  of  the  7  equal  parts  of  35  ? 

7.  When  you  find  how  many  times  9  cents  are  contained 
in  72  cents,  is  the  result  a  concrete  or  an  abstract  number  ? 

8.  When  you  find  one  of  the  9  equal  parts  of  72  cents,  is 
the  result  a  concrete  or  an  abstract  number  ? 

9.  How  many  times  are  9  dollars  contained  in  29  dollars, 
and  how  many  dollars  remain  ? 

4  * 


42  DIVISION. 

10.  If  you  have  47  peaches  to  distribute  among  7  boys,  how 
many  entii'e  peaches  can  you  give  to  each,  and  how  many 
peaches  will  remain  ? 

11.  7  times  6  peaches,  and  5  peaches,  are  how  many  peaches  ? 

DEFINITIONS. 

78.  Division  is  the  process  of  finding  how  many  times  one 
number  is  contained  in  another ;  or. 

Division  is  the  process  of  separating  one  of  two  numbers  into 
as  many  equal  jiarts  as  there  are  units  in  the  other. 

79.  The  Dividend  is  the  number  to  be  divided. 

80.  The  Divisor  is  the  number  by  which  to  divide. 

81.  The  Quotient  is  the  result  obtained  by  the  division. 

82.  The  Remainder  is  a  part  of  the  dividend  remaining  un- 
divided. 

83.  The  Si^  of  Division  is  -=-,  and  is  read,  divided  hi/.  The 
dividend  is  placed  at  the  left  of  the  sign,  and  the  divisor  at 
the  right  of  it. 

Thus,  40  -H  8  is  read,  forty  divided  by  eight. 

Division  is  sometimes  denoted  by  placing  the  dividend  over 
the  divisor,  with  a  line  between  them. 

Thus,  ^;^  is  read,  sixteen  divided  by  four. 

Division  is  also  denoted  by  a  curved  line,  ),  with  the  divisor 
on  the  left  and  the  dividend  on  the  right. 

Thus,  5)10  is  read,  ten  divided  by  five. 

84.  A  Parenthesis  (  ),  enclosing  two  or  more  numbers,  or  a 
Vinculnm,  ,  drawn  over  them,  denotes  that  the  expression 
is  to  be  treated  as  a  single  number. 

Thus,  (16  +  4)  -f-  5,  or  li5^f^-4-  5,  denotes  that  the  sum  of  16  and  4, 
or  20,  is  to  be  divided  by  5. 

85.  The  Xamcs  of  the  equal  parts  into  which  a  number  may 
be  divided  differ  according  to  their  number. 

A  half  of  a  number  is  one  of  two  equal  parts  into  which  it 
is  divided. 


DIVISION.  43 

A  third  of  a  number  is  one  of  the  three  equal  parts  into 
which  it  is  divided. 

A  fourth  of  a  number  is  one  of  the  Jour  equal  parts  into 
which  it  is  divided. 

In  like  manner  we  have  the  names  fifths,  sixths,  sevenths, 
eighths,  ninths,  tenths,  twentieths,  thirti/ fourths,  forty-sixths,  etc. 

Halves,  thirds,  fourths,  etc.,  are  expressed  by  writing  the 
number  denoting  the  name  of  the  parts,  as  a  divisor,  under  a 
line,  and  the  number  denoting  the  number  of  the  parts  repre- 
sented, as  a  dividend,  above  the  line. 

Thus,  ^  signifies  1  divided  by  2,  or  1  half  of  1,  and  is  read,  one  half. 
f  signifies  2  divided  by  3,  or  2  thirds  of  1,  and  is  read,  two  thirds. 

86.  A  Fraction  is  a  number  which  represents  one  or  more 
of  the  equal  parts  into  which  a  unit,  or  one,  is  divided. 
Tims,  I,  expressing  3  of  the  four  equal  parts  of  1,  is  a  fraction. 

Principles  of  Division. 

87. — 1.  Division  is  the  reverse  of  multiplication. 

2.  The  quotient  must  he  an  abstract  number  when  the  divisor 
and  dividend  are  similar  numbers. 

3.  The  quotient  must  be  a  concrete  number  and  the  divisor  an 
abstract  number  when  the  divisor  and  dividend  are  dissimilar 
numbers. 

4.  The  remainder  and  dividend  must  be  similar  mimbers. 

5.  The  dividend  is  equal  to  the  product  of  the  integer  of  the 
quotient  multiplied  by  the  divisor,  jylus  the  remainder. 

ca.se;  I. 
Short  Division. 

88.— Ex.  1.  Divide  4313  by  4. 

Divisor,  4^)4-313     Dividend.  Solution.— For  convenience  we 

.          i             .  first  divide  the  liighest   order   of 

107 8 J  Quotient.  ^^itg^     4  jg  contained  in  4  thcu- 

4  sands  1  thousand  times.     Write  1 

, (D-i a      Tk       I-  i"    the    thousands'    order    in    the 

4-313    Proof  q^^^^j^^^ 

4   is   not  contained   in   3   hundreds  any  number  of  hundred  times. 


44  DIVISION. 

Write  0  in  the  huiulrods'  order  in  the  quotient,  Jind  unite  the  3  hundreds 
with  the  1  ten,  making  31  tens. 

4  is  contained  in  31  tens  7  tens  times,  with  a  remainder  3  tens.  "Write 
7  in  the  tens'  order  in  tlie  quotient,  and  unite  the  3  tens  with  tlie  3  ones, 
making  33  ones. 

■  4  is  contained  in  33  ones  8  times,  with  a  remainder  1.  Write  8  in  the 
ones'  order  in  tlie  quotient,  and  the  remainder  1,  with  4,  the  divisor, 
under  it  as  a  part  of  the  quotient.     The  required  quotient  is  107S|. 

Prove  the  correctness  of  the  solution  by  multiplying  the  integer  of  the 
quotient  by  the  divisor,  and  adding  the  remainder.  For,  (Art.  87 — oj 
the  dividend  must  be  equal  to  the  integer  of  tlie  quotient  multiplied  by 
the  divisor,  plus  the  remainder. 

The  explanation  of  the  solution  may  be  abridged  thus :  4  in  4,  1 ;  4 
in  3,  0;  4  in  31,  7 ;  4  in  33,  8,  with  1  as  a  remainder.  Ans.  1078|^. 

Division  is  called  Short  Division  when  in  the  solution  only 
the  divisor,  dividend  and  quotient  are  written. 

Solve  and  prove — 

(2.)  (3.)  (4.)  (5.)  (6.) 

4)940         6)672  5)6570  7)847         8)968 

(7.)  (8.)  (9.)  (10.)  (11.) 

8)4162        2)1931        6)6753        5)47235       9)817 

12.  If  6  boys  share  equally  1386  apples,  liow  many  -will 
each  boy  have?  Ans.  231. 

C^SE    II. 

Long  Division. 

89.— Ex.  1.  Divide  16013  by  5,  or  find  one  fifth  of  16013. 

Divisor.    Dividend.     Quotiont.  ^,  -  •  .  ^    •       i    •      i    , 

_j  Solution. — 6  is  not  contained  in  1  ten- 

OyluUlo(o/C(Jr.^-        thousand     any    number    of    ten-thousand 

^Q  times ;  hence,  we  unite  the  1  ten-thousand 

with  the  6  thousands,  making  16  thousands. 

5  is  contained  in  16  thousands  3  thou- 

sands  times,  with  a  remainder.     Write  3  in 

]^g  the  thousands'  order  of  the  quotient.    5X3 

■tQ  thousands  -=  15   thousands,  which,  written 

iHider    the    16    tiiousands   and    subtracted, 

3  Tirm.  leaves  1  tiiousand.     Unite  the  1   thousand 

with  the  0  hundreds,  making  10  hundreds. 


10 
10 


DIVISION. 


45 


6  is  contained  in  10  hundreds  2  hundred  tiraos.  Write  2  in  the  hun- 
dreds' order  of  the  quotient.  5X2  hundreds  =-  10  liundreds,  wliich, 
written  under  the  10  liundreds  and  subtracted,  leaves  no  remainder. 

0  is  not  contained  in  1  ten  any  number  of  tens  times.  Write  0  in  the 
tens'  order  of  the  quotient,  and  unite  with  the  1  ten  the  3  ones,  making 
13  ones. 

5  is  contained  in  13  ones  2  ones  times,  with  a  remainder.  Write  2  in 
the  ones'  order  in  the  quotient.  5X2  ones  =  10  ones,  which,  written 
under  the  13  ones  and  subtracted,  leaves  3  remainder. 

W^rite  the  remainder  3  over  the  divisor  5  as  a  part  of  the  quotient, 
which  gives  3202|,  the  result  required. 


3202-- 

5 


16010 

3 

16013 


Proof. — Prove  the  solution  by  multiplying  the  integer 
of  the  quotient  by  the  divisor,  and  adding  the  remainder, 
which  gives  the  dividend ;  hence,  the  work  is  correct. 


Division  is  called  Long  Division  when  each  process  of  the 
solutiou  is  written. 


Solve  and  prove — 

(2.)  (3.) 

8)3368(  1 1)235  4-( 

(6.)  (7.) 

21)640(  15)915( 


(4.) 
9)3706( 

(8.) 

25)806( 


(5.) 
12M004( 

(9.) 

7)1803( 
Am.  220f. 


10.  Find  one  eighth  of  1765. 

11.  When  a  steamer  sails  3018  miles  in  14  days,  what  is  her 
progress  per  day  ?  A^is.  21 5 j\  miles, 

12.  At  the  rate  of  27  miles  per  hour,  in  how  many  hours 
will  a  train  of  cars  run  2563  miles?  Ans.  94ff . 


C^SE    III. 

When  the  Divisor  is  1,  with  One  or  More  Ciphers  on  the  Rig-ht. 

90.  It  has  been  shoAvn  that  a  number  is  multiplied  by  10 
by  removing  each  figure  one  order  to  the  left,  which  is  done  by 
annexing  one  cipher ;  by  100,  by  removing  each  figure  two 
orders  to  the  left,  or  annexing  txvo  ciphers,  etc.    (Art.   74.) 


4(i  DIVISION. 

Hence,  since  division  is  the  reverse  of  multiplication  (Art.  87), 

To  divide  by  10,  remove  each  figure  one  order  to  the  right  by 
removing  the  decimal  point  one  order  to  the  left. 

To  divide  by  100,  remove  each  figure  two  orders  to  the  right 
by  removing  the  decimal  point  two  orders  to  the  left,  etc.     Thus, 

2563  -=-  10  is  256.3,  or  256yV 
2563  --  100  is  25.63,  or  2b^i^. 
2563  --  1000  is  2.563,  or  2^W 

The  decimal  point  in  the  result  separates  the  integer  of  the 
quotient  from  the  fractional  part. 

The  first  order  on  the  right  of  the  decimal  point  is  tenths ;  the 
second  order,  hundredths ;  the  third,  thousandths. 

Ex.  1.  Divide  67325  by  1000. 

67325  -4-  1000  =  Solution. — Remove  the  decimal  point  in 

/?'y  <qop:        r"v  ^^^  ^^^^  dividend  three  orders  to  the  left,  giving 

6  /.J^O  =  6  /-^^  67.325,  or  67xW(j,  tlie  quotient  required. 


2.  Divide  36540  by  100.  Ans.  365^ 


iJL 
00- 


3.  Divide  4632  by  10.  Am.  4631^^. 

4.  How  many  thousands  in  5328  ?  Ans.  ^ywm: 

5.  How  many  thousand  feet,  and  how  many  feet  over  exact 
thousands,  are  1634500  feet  of  boards? 

Ans.  1634  thousand  feet,  and  500  feet  over. 

6.  What  is  the  quotient  of  31638954  divided  by  100000  ? 

91.  Rule  for  Division.— Z)z*z;7:^(5  the  least  number  of  the 
left-hand  order  a  of  the  dividend  that  will  contain  the 
divisor,  and  place  the  quotient  at  the  right  of  the 
dividend  in  long  division,  and  beneath  the  dividend 
in  short  division. 

Multiply  the  divisor  hy  this  quotient;  subtract  the 
result  from  that  part  of  the  dividend  which  ivas  used; 
to  the  rrinaindycr  anne.v  the  ne,vt  order  of  the  d,ivi- 
dend,  and  divide  the  number  thus  formed.  Proceed 
in  like  manner  until  all  the  dividend  has  been  used. 


DIVISION. 


47 


If  there  he  at  last  a  remainder,  w?'ite  it,  with  tJie 
divisor  xmder  it,  as  a  fractional  part  of  the  quotient. 

When  the  divisor  is  1,  with  one  or  more  ciphers  on 
the  ri<3ht,  remove  the  deciiyial  point  in  the  dividend  as 
many  orders  to  the  left  as  there  are  ciphers  on  the 
right  of  the  divisor.  The  orders  on  the  left  of  the 
point  will  be  the  integer  of  the  quotient,  and  the 
orders  on  the  right,  the  fractional  part  of  it. 

PROOF.— Multiply  the  integer  of  the  quotient  by  the 
divisor,  and  add  to  the  product  the  remainder,  if  any. 
If  the  worlc  is  correct,  this  result  will  equal  the  divi- 
dend. 


Divide — 

1.  74317  by  6. 

2.  10570  by  35. 

3.  40161  by  100. 

4.  18312  by  24. 

5.  99031  by  62. 

6.  13505  by  37. 

7.  47256  by  6. 


PROBLEMS. 


Ans. 


12386f 


Ans.  302. 


Am.  763. 


Ans.  365. 


Ans.  3199. 
Ans.  1002. 


Ans.  97^. 


Ans.  2750. 


How  many  times — 


15. 
16. 
17. 
18. 
19. 


21  in  1223? 
43  in  34165? 
18  in  1499  ? 

15  in  75850? 

16  in  3251  ? 


Ans. 


58^. 


Ans.  83^^. 
Ans.  203^. 


8.  31990  by  10. 

9.  71142  by  71. 

10.  33443  by  8. 

11.  2815  by  29. 

12.  4449  by  101. 

13.  68750  by  25. 

14.  42122  by  103. 

How  much  is — 

20.  One  fifth  of  95340  ? 

21.  One  tenth  of  9534? 

22.  One  eighty-first  of  973  ? 

23.  One  nineteenth  of  3640? 

24.  One  thousandth  of  31673  ? 


25.  When  112  muskets  are  worth  1344  dollars,  what  is  the 
value  of  each?  Ans.  12  dollars. 

26.  How  many  tons  of  coal,  at  10  dollars  per  ton,  can  be 
bought  with  13670  dollars? 

27.  65120  --  37  =-  what  number?  Ans.  1760. 

28.  1651  ^  127  =  what  number? 

29.  14150 --115  =  what  number?  Ans.  1^12,^. 

30.  13200^-48==  what  number? 

31.  208126 --345  =  what  number?  Ans.  603^. 


48  DIVISION. 

32.  How  many  acres  of  land  at  100  dollars  each  can  be 
bought  for  25605  dollars  ? 

33.  A  field  of  101  acres  yields  2125  bushels  of  grain.  How 
much  is  the  yield  per  acre  ?  Ans.  21yi j  bushels. 

34.  A  man  has  14250  dollars,  Avhich  he  wishes  to  invest  in 
horses  at  250  dollars  each.    Hov>'  many  horses  can  he  purchase? 

35.  It  is  proposed  to  divide  a  ti'act  of  land  containing  72000 
acres  into  farms  of  320  acres  each.  How  many  farms  will  it 
make?  Ans.  225. 

36.  In  an  orchard  there  are  44520  trees  in  212  equal  rows. 
How  many  trees  are  there  in  each  row  ? 

37.  If  the  dividend  is  126072  and  the  divisor  612,  what  is 
the  quotient  ? 

38.  The  height  of  a  mountain  in  Asia  is  28176  feet,  and  the 
height  of  Mount  Washington  is  6284  feet.  How  mauy  times 
as  high  as  the  latter  is  the  former?  Ans.  4|-|^. 


TEST    QUESTIONS. 

92. — 1.  What  is  Division?  The  dividend?  The  divisor?  The 
quotient  ?  "What  in  division  corresponds  to  the  factors  of  tlic  product  in 
muhiplication ?  What  is  the  product?  What,  then,  in  division  maybe 
regarded  as  the  factors  of  the  dividend? 

2.  What  is  the  Sign  of  division  ?  When  written  between  two  num- 
bers what  does  it  denote?     In  what  other  ways*  may  division  be  denoted  '/ 

3.  Recite  the  Principles  of  division.  Sliow  that  division  is  the  re- 
verse of  multiplication.  Show  when  the  quotient  will  be  an  abstract 
number.     When  the  quotient  will  be  a  concrete  number. 

4.  What  is  a  Rkmainder  in  division  ?  Why  are  the  remainder  and 
dividend  similar  numbers?  How  may  the  remainder  be  changed  to  a 
fractional  part  of  the  quotient  ? 

5.  Recite  tlie  Rule  for  division.  How  does  long  division  clifier  from 
short  division  ?  What  is  the  reason  for  removing  the  decimal  point  to 
the  left  in  dividing  by  10,  100,  etc.  ? 

0.  What  is  the  Pkoof  of  division  ?  The  reason  for  it  ?  Since  multi- 
plication and  division  are  the  reverse  of  each  other,  how  may  multipli- 
cation be  proved  ?  Sliow  that  the  product  divided  by  the  multiplier 
gives  the  multiplicand. 


REVIEW.  49 

SECTION    VIII. 
REVIEW  PROBLEMS. 

MENTAL   EXERCISES. 

93.— Ex.  1.  The  factors  of  a  product  are  11  and  12.  What 
is  the  product? 

2.  The  product  of  15  by  9  is  what  number? 

3.  If  a  boy  can  earn  18  dollars  in  one  month,  how  many 
dollars  can  he  earn  in  10  months?  • 

4.  At  14  dollars  a  ton,  what  will  13  tons  of  hay  cost? 

Solution. — If  one  ton  cost  14  dollars,  13  tons  will  cost  13  times  14 
dollars,  or  182  dollars.  The  multiplication  may  be  conveniently  per- 
formed thus :  13  times  14  =  3  times  14  +  10  times  14 ;  3  times  14  =  42 ; 
10  times  14  =  140 ;  42  +  140  =  182. 

5.  At  16  dollars  each,  what  will  12  garments  cost? 

6.  If  14  men  can  do  a  piece  of  work  in  21  days,  in  how 
many  days  can  one  man  do  it  ? 

7.  John  lives  5  miles  from  a  certain  place ;  Andrew,  7  miles 
farther  away ;  and  Benjamin,  8  times  as  far  as  Andrew.  How 
far  from  the  place  does  Benjamin  live  ? 

8.  How  many  lengths  of  12  feet  each  are  there  in  a  fence 
which  is  132  feet  long? 

9.  A  and  B  start  from  points  121  miles  apart,  and  travel 
toward  each  other,  A  at  the  rate  of  5  miles  per  hour,  and  B  at 
the  rate  of  6  miles  per  hour.    In  how  many  hours  will  they  meet  ? 

10.  When  flour  is  worth  8  dollars  a  barrel,  how  many 
barrels  of  flour  will  pay  for  24  tons,  of  coal  at  5  dollars  a  ton  ? 

11.  If  10  men  can  do  a  piece  of  work  in  6  days,  in  how 
many  days  can  4  men  do  it  ? 

Solution. — If  10  men  can  do  a  piece  of  work  in  6  days,  1  man  can 
do  it  in  10  times  6  days,  which  are  60  days ;  and  4  men  can  do  it  in  one 
fourth  of  60  days,  which  is  15  days. 

12.  When  12  tons  of  coal  at  7  dollars  a  ton  pay  for  21 
pairs  of  boots,  how  much  are  the  boots  worth  a  pair  ? 

13.  If  8  cords  of  wood  will  buy  32  pairs  of  shoes,  how  many 
cords  will  buy  40  pairs  of  shoes  ? 


50 


BEri£!W. 


14,  I  sold  5  dozen  of  eggs  at  the  rate  of  4  for  7  cents,  and 
received  5  cents  in  money,  and  the  balance  in  coffee  at  25  cents 
a  pound.     How  many  pounds  of  coffee  did  I  receive  ? 


WMITTEN  EXERCISES. 

94. — Ex.  1.  If  in  making  mortar,  6  bushels  of  sand  are  re- 
quired for  each  cask  of  lime  used,  how  many  bushels  of  sand 
will  be  required  for  97  casks  of  lime  ?  Ans.  582. 

2.  How  many  loads,  each  contaiuiug  1345  bricks,  are  there 
in  a  pile  containing  286485  bricks  ? 

3.  If  the  front  and  rear  walls  of  a  house  each  require  for 
their  construction  31G50  bricks,  and  the  other  two  walls  each 
require  43400  bricks,  how  many  bricks  are  required  for  the 
four  walls?  '  Aiis.  150100. 

4.  The  factors  of  a  product  are  8043  and  405.  What  is 
that  product?  Ahs.  1232415. 

5.  The  product  of  two  factors  is  9225 ;  onfe  of  the  factors 
is  45.     What  is  the  other  factor? 


45)9225(205 
90_ 

"225 
225 


PoM'TioN. — Since  a  product  is  the  result  ob- 
tained by  multiplying  one  of  two  factors  by  the 
otlier,  the  quotient  obtained  l)y  dividing  the  prod- 
uct by  one  of  the  factors  must  be  the  other  factor. 
9225  -f-  45  ^  205,  the  factor  required. 


HE  VIEW.  61 

6.  If  a  product  is  9225  and  the  multiplicand  205,  what  is 
the  multiplier? 

7.  If  a  man  has  15000  dollars,  and  should  spend  enough  of 
it  to  pay  for  a  farm  of  80  acres  at  78  dollars  per  acre,  how 
much  would  he  have  left  ? 

8.  If  a  merchant  should  purchase  1011  barrels  of  flour  at 
12  dollars  a  barrel,  and  pay  down  7919  dollars,  how  much 
would  he  then  owe  for  the  flour?  A^is.  4213  dollars. 

9.  The  dividend  is  15750  and  the  divisor  25.  What  is  the 
quotient,  or  the  other  factor  of  the  dividend  ?  Aiis.  630. 

10.  What  is  the  average  of  16,  22  and  28? 

Solution. — The  average  of  two  numbers  is  one  half  of  the  sum  of 
those  numbers ;  the  average  of  th7-ee  numbers  is  one  third  of  their  sum, 
etc.     The  sum  of  16,  22  and  28  is  66,  and  one  third  of  66  is  22. 

11.  The  elevation  of  the  northern  lakes  above  the  sea  is  as 
follows :  Superior,  627  feet ;  Michigan,  587  feet ;  Huron,  574 
feet ;  and  Ontario,  282  feet.  What  is  their  average  elevation 
above  the  sea?  Ans.  517|- feet. 

12.  What  is  the  value  of  (14  +  6)  +  16  X  2  —  (^63  — 
19~+4)  +  45  ^  9  —  3  ? 


iU  +  6')  +  16X,?—  (63—19  +  4)  +45-^9  —  3 

=  20+  16  X  !2  —  40  +  45^9  —  3 

=  20  +  32  —  40  +  5—3;  =57-43;  =14 

Solution. — Combining  first  the  numbers  in  the  parentheses,  we  have 
20  for  the  value  of  (14  +  6),  and  40  for  the  vahie  of  (63  — r9  +  4). 
Combining  the  numbers  affected  by  the  signs  of  multiplication  and  those 
by  the  sign  of  division,  we  have  32  for  the  value  of  16  X  2,  and  5  for 
45  -=-  9.  Then,  combining  the  numbers  as  indicated  by  the  signs  of  ad- 
dition and  subtraction,  we  have  14  as  the  value  required. 

13.  What  is  the  value  of  4  +  6  X  5  —  16  --  8  —  (4  X  2)  ? 

14.  Henry  has  7375  dollars,  which  is  7  times  as  much  as 
Daniel  has,  lacking  780  dollars.  How  many  dollars  has 
Daniel?  Ans.  1165. 

15.  Tv.o  candidates  at  an  election  received  in  the  aggregate 
15653  votes.  If  one  of  them  received  783  votes  more  than  the 
other,  how  manv  votes  did  the  other  receive  ? 


52  '  FACTORING. 

SECTION   IX. 
FACTORS  AKD   DIVISORS. 

95. — Ex.  1.  Of  what  two  integers  is  6  the  product  ? 

2.  What  two   integers   multiplied    together    produce    15? 
What  two  produce  21  ? 

3.  What  integers  are  factors  of  6  ?     Of  15  ?     Of  21  ? 

4.  Of  what  three  integers  is  30  the  continued  product  ? 

5.  Of  what  sets  of  two  integers  is  30  the  product  ? 

30  =  2  X  15,  or  3  X  10,  or  5  X  6. 

6.  Of  what  sets  of  integers  greater  than  1  is  24  the  product  ? 

7.  Name  some  numbers  that  are  the   product    of  integers 
greater  than  1. 

8.  What  are  the  smallest  integers  greater  than  1  that  will 
divide  21  without  a  remainder  ?     30  without  a  remainder  ? 

9.  Give  the  sets  of   integers  greater  than   1  which,  when 
multiplied  together,  will  produce  30. 

10.  Of  what  number  are  3  and  5  the  factors?     2,  3  and  5 
the  factors  ? 

DEFINITIONS. 

96.  The  Factors  of  a  number  are  the  integers  which  being 
multiplied  together  will  produce  that  number. 

Thus,  2,  3  and  5  are  the  factors  of  30 ;  for  2  X  3  X  5  =  30. 

97.  A  Prime  Number  is  an  integer  that  has  no  factor  except 
itself  and  1. 

Thus,  1,  3,  5,  7  and  11  are  prime  numbers. 

98.  A  Composite  Number  is  an  integer  that  has  other  fac- 
tors besides  itself  and  1. 

Thus,  4  and  6  are  composite  numbers,  since  4  =  2X2;  and  6  =:  2  X  3. 

99.  A  Prime  Factor  is  a  factor  which  is  a  prime  number. 
The  prime  factor  1  is  not  commonly  mentioned,  since  it  is  a  factor  of 

every  integer. 

Numbers  are  said  to  be  vtutiially  prime,  or  prime  to  each  other,  when  they 
have  no  common  factor  exce«t  1. 


FACTORING.  58 

100.  Factoring  is  the  process  of  finding  the  factors  of  com- 
posite numbers. 

The  number  of  times  a  number  is  taken  as  a  factor  may  be  denoted  by 
writing  a  small  figure,  called  an  Exponent,  at  the  right  and  above  the 
figure  or  figures  of  the  factor. 

Thus,  3^  =;  3  X  3,  and  denotes  that  3  is  taken  twice  as  a  factor. 

11^  =  11  X  11  X  11,  iind  denotes  that  11  is  taken  3  times  as  a  factor. 

101.  An  Exact  Divisor  of  a  number  is  any  integer  which 
will  divide  the  number  without  a  remainder. 

Thus,  1,  2,  3,  4,  G  and  12  are  each  exact  divisors  of  12. 
The  Exact  Divisors  of  a  number  are  called,  also.  Divisors,  or  Measures, 
of  that  number,  and  must  be  factors  of  it. 

A  number  is  said  to  be  divisible  by  its  exact  divisors. 

Thus,  12  is  divisible  by  its  exact  divisors  1,  2,  3,  4,  6  and  12. 

102.  Any  number  is  divisible  by  2  when  its  right-hand' 
figure  is  0,  2,  4,  6,  or  8. 

For  such  numbers  are  composed  of  some  exact  number  of  twos. 

Numbers  divisible  by  2  are  Even  Numbers,  and  all  others 
are  Odd  Numbers. 

103.  A  number  is  divisible  by  4  if  its  tens  and  ones  are 
divisible  by  4. 

For  4  is  an  exact  divisor  of  100,  and  of  any  number  of  hundreds; 
hence,  if  the  tens  and  ones  of  a  number  are  divisible  by  4,  the  number 
itself  must  be. 

Thus,  648  and  7312  are  each  divisible  by  4. 

104.  A  number  is  divisible  by  5  if  its  right-hand  figure  is 
0  or  5. 

A  number  whose  right-hand  figure  is  0  is  an  exact  number  of  tens; 
a  number  whose  right-hand  figure  is  5  is  an  exact  number  of  tens  plus  5. 
5  is  an  exact  divisor  of  5  or  10 ;  hence,  any  exact  number  of  tens,  or  any 
exact  number  of  tens  plus  5,  is  divisible  by  5. 

Thus,  70  and  75  are  each  divisible  by  5. 

105.  A  number  is  divisible  by  3  or  9  when  the  sum  of  the 
ones  represented  by  its  figures  is  divisible  by  3  or  9. 

Take,  for  example,  the  number  7542  ;  7542  =  7000  +  500  +  40  +  2 ;  and 
7000  =  7  times  999  +  7  ;  500  =  5  times  99  +  5 ;  40  =  4  times  9  +  4 ; 
and  2  =  2;  where  the  figures  expressing  the  number  of  each  order  plus 

5* 


54  FACTORING. 

the  exact  number  of  times  9  are  the  figures  of  the  given  number.     Now, 
7  times  999,  5  times  99  and  4  times  9,  being  each  divisible  by  9  and  by  3, 
if  the  sum  of  the  ones  represented  by  the  figures  of  the  number  are 
divisible  by  9  or  by  3,  the  number  itself  is  thus  divisible. 
Thus,  7542  and  9765  are  each  divisible  by  9  and  by  3. 

106.  Principles. — 1.  Every  number  is  equal  to  the  product 
of  alt  iLs  prime  factors. 

2.  A  number  is  divisible  by  all  its  j^rime  factors,  and  by  ah- 
the  products  of  two  or  more  of  them,  and  is  divisible  by  no  other 
numbers. 

WRITTEN  EXERCISES. 

10?. — Ex.  1.  What  are  the  prime  factors  of  70? 

Solution.  —  Since    the     right-hand 

-^^^  figure  is  0,  we  can  divide  by  the  prime 

g)QQ  numbers  2  and  5.  (Arts.  102  and  104.) 

Dividing    by   these    prime    numbers 

gives   for   a  quotient  7,   which  is   also 

Proof,  J?  X  <5  X  7  =  70       prime.     Hence,  the  prime  factors  of  70 

are  2,  5  and  7. 


7 


2.  What  are  all  the  factors  or  divisors  of  66  ? 

2)66  2y  3,  11  Solution.— Since  every  prime 

Q)'^g  S  y^  2  =     6       factor  of  a  number,  and  every 

—  -ii  ^  ^  =  $>^        product  of  two  or  more  of  these 

^-'-  1  1  \(  Q '9'??        prime  factors,  is  an  exact  divisor 

of   the   number,   and    no   other 
11  /\  o  y^  /C  =  ob        numbers  can  be  exact  divisors  of 
tliat  number  (Art.  106 — 2),  the  prime  factors  2,  3  and  11,  and  the  prod- 
ucts of  3  by  2,  11  by  2,  11  by  3,  and  11  by  2  times  3,  must  be  all  the  fac- 
tors or  exact  divisors  of  66. 

Hence,  2,  3,  11,  6,  22,  33  and  66  ar£  the  factors  and  divisors  required. 

3.  What  are  the  prime  factors  of  84?      Av.<<.  2\  -S  and  7. 

4.  What  are  all  the  factors  or  divisors  of  56  ? 

Ans.  2\  7,  4,  8,  14,  28,  56. 

108.  Rule  for  Factoring.— D/f^fZe  the  given  ninnber  hy 
any  of  its  prim,e  factors  greater  than  1.  Divide  the 
quotient,  if  composite,  in  lilce  vnanner,  and  so  proceed 


FA  CTORING. 


55 


jontil  the  last  quotient  is  a,  priDie  iiiunhcr.  The  last 
quotient  and  the  several  divisors  will  be  the  prime 
factors  of  tJie  ninnber. 

Tlie  prime  factors  of  a  given  number,  and  the 
various  products  of  these  factors,  by  talcing  two 
together,  three  together,  etc.,  are  all  the  different  fac- 
tors or  exact  divisors  of  that  number. 

1^  no  JUL  EMS. 

What  are  the  prime  factors  of — 

Ans.  3^  11. 
Ans.  3,  7,  11. 

Ans.  2,  3,  5,  37. 

Ans.  3,  7,  17,  19. 

What  are  all  the  different  factors  or  divisors  of — 
13.  70?      Ans.  2,  5,  7,  10,14,    16.  42?       Ans.  2,  3,  fi,  7,  14, 

21  and  42. 
17.  63? 

18.105?    ^n.s.  3,  5,  7,  15,  21, 
35  and  105. 


1. 

75? 

Avs.   3,  5,  5. 

7. 

99? 

2. 

144? 

Ans.   2^  3^ 

8. 

231? 

3. 

116? 

9. 

875? 

4. 

340? 

Ans.  2\  5,  17. 

10. 

1110? 

5. 

180? 

11. 

4004? 

6. 

3809  ? 

Ans.   13,  293. 

12. 

6783? 

35  and  70. 

14.  30? 

15.  56?    Ans.  2,4,7,8,14,28 

and  56. 


19.  How  many  of  the  different  factors  or  divisors  of  100  are 
prime,  and  how  many  are  composite  ? 

Ans.  Three  are  prime  and  gix  are  composite. 

20.  How  many  of  the  different  factors  or  divisors  of  210  are 
prime,  and  how  many  are  composite  ? 


TEST    QUESTTONS. 

109. — 1.  What  are  the  Factors  of  a  number?  What  is  a  prime 
number?  Name  some  number  that  is  the  product  of  two  prime  num- 
bers. What  is  a  composite  number  ?  Give  an  example  of  a  composite 
number. 

2.  Wliat  is  a  Prime  Factor  ?  Name  a  prime  factor  of  30.  A  com- 
posite factor  of  30.     When  are  two  numbers  prime  to  each  other? 


56  FA  CTORING. 

3.  What  is  an  Exact  Divisor  of  a  number  ?  How  do  even  numbers 
differ  from  odd  numbers  ?  Of  what  are  divisors  or  measures  of  a  num- 
ber factors  ? 

4.  By  what  is  a  number  said  to  be  Divisible  ?  How  can  you  know 
that  a  number  is  divisible  by  2  ?  That  a  number  is  divisible  by  4  ?  By 
5?     By  9  or  3? 

5.  What  is  Factoring  ?  How  do  you  denote  the  number  of  times  a 
factor  is  taken  ?     How  do  you  find  all  the  factors  of  a  number  ? 

6.  What  are  Principles  of  factoring?  Show  that  18  is  the  product 
of  all  its  prime  factors.  Show  that  18  is  divisible  by  the  various  prod- 
ucts of  its  prime  factors. 

7.  What  is  the  Rule  for  factoring  ?  In  factoring  an  even  number 
why  do  you  divide  by  2  ?  In  factoring  numbers  why  do  you  divide  by 
their  prime  factors  in  succession  ? 


SECTION   X. 

commojY  divisors. 

110. — Ex.  1.    What   numbers   are   exact   divisors   of    15? 
Of  20? 

2.  What  number  is  an  exact  divisor  of  both  15  and  20? 

3.  What  is  the  unit  of  15?     Of  20?     Of  the  exact  divisor 
of  both  15  and  20? 

4.  What  numbers  are  exact  divisors  of  33  dollars  ? 

5.  What  number  is  an  exact  divisor  of  both  22  dollars  and 
33  dollars  ? 

6.  What  is  the  unit  of  22  dollars  ?     Of  33  dollars  ?     Of  the 
exact  divisors  of  both  22  dollars  and  33  dollars  ? 

7.  What  are  the  exact  divisors  common  to  12  and  18? 

8.  What  prime  factors  have  12  and  18  in  common?     ^Yliat 
is  the  product  of  those  factors  ? 

{).   What  is  the  greatest  exact  divisor  common  to  12  and  18? 
To  8  and  20? 

10.  What  exact  divisor  is  common  to  5  and  3  times  5? 

11.  What  exact  divisors  arc  common  to  G  and  7  times  6? 

12.  AVhat  exact  divisors  are  common  to  0  and  42  ?    To  6 
and  42  and  their  sum?     To  6  and  42  and  their  difference? 


FACTORING.  57 

DEFINITIONS. 

111.  A  Common  Dirisor,  or  Commou  Measure,  of  two  or  more 
numbers  is  any  exact  divisor  (Art.  101)  of  each  of  tho.se 
numbers. 

Thus,  2  is  a  common  divisor  of  8,  10  and  18. 

Only  similar  numbers  can  have  a  common  divisor,  since  one 
number  to  measure  another  must  have  the  same  unit. 

112.  The  Greatest  Common  Divisor,  or  Greatest  Common 
Measure,  of  two  or  more  numbers  is  the  greatest  exact  divisor 
of  each  of  them. 

Thus,  6  is  the  greatest  common  divisor  of  12,  24  and  42. 

Numbers  that  are  prime  to  each  other  have  no  common 
divisor,  for  they  can  have  no  common  factor  greater  than  1. 

113.  Principles. — 1.  The  greatest  common  divisor  of  two  or 
onore  numbers  is  the  product  of  all  their  common  prime  factors. 

2.  A  divisor  of  a  number  is  a  divisor  of  any  integral  number 
of  times  that  number. 

3.  A  common  divisor  of  two  or  more  numbers  is  also  a  divisor 
of  their  sum  and  of  their  difference. 

C^SE   I. 
Common  Divisor. 

114. — Ex.  1.  What  is  a  common  divisor  of  14  and  20. 

Solution. — 2  is  an  exact  divisor  of  any  number  whose 
^Jj-A-i  '^^       right-hand  figure  is  4  or  0  (Art.  102);  hence  2  is  an 
2'    ]^Q       exact  divisor  of  both  14  and  20,  and  therefore  a  com- 
mon divisor  of  the  given  numbers. 

2.  What  are  all  the  common  divisors  of  45  and  90  ? 

45  =  3^3X5  3)45,  90  Solution.  — By  fac- 

90=^3X3X5X2    ^r      '?)7^     '^n      Coring  we  find  the  com- 
or,    oyio,  ou      ^^^   ^^-^^^  factors  of 

9  x^  o-_     n                     5j    5,  10      4:5  and   90   to  be  3,    3 

\,                           ~~i  ^      and  5 ;  lience  3, 3  and  5 

'  are  common  divisors  of 

3X3X5  =  45  45and90. 


58  FACTORING. 

Since  every  product  of  two  or  more  of  the  prime  factors  must  be  a 
divisor,  the  various  products  that  can  be  formed  by  3,  3  and  5  must 
likewise  be  common  divisors  of  the  given  numbers.  These  products  are 
3  X  3,  or  9 ;  5  X  3,  or  15 ;  and  5  X  3  X  3,  or  45.  Hence,  3,  5,  9,  15 
and  45  are  the  common  divisors  required. 

3.  What  is  a  common  divisor  of  27  aud  39?  Ans.  3. 

4.  What  are  all  the  common  divisors  of  45  and  75  ? 

Ans.  3,  5  and  15. 
115.  Rule  for  finding  Common  Divisors.— i^/^cZ  tlie  prime  fac- 
tors of  the  given  nwrvibers.     All  coimnon  factors  are 
eoTmyion  divisors,  and  all   the  various  products  of 
those  factors  are  all  the  coininon  divisors. 

PROBLEMS. 

1.  What  is  a  common  divisor  of  25  and  35?  Ans.  5. 

2.  What  are  all  the  different  common  divisors  of  27  and  81  ? 

3.  What  are  the  common  composite  factors  or  divisors  of  24, 
72  and  84?  J ws.  4,  6  and  12. 

4.  What  are  all  the  common  divisors  of  105,  210,  315. 

Arts.  3,  5,  7,  15,  21,  35  and  105. 

C^SK    II. 

Greatest  Common  Divisors. 

116. — Ex.  1.  What  is  the  greatest  common  divisor  of  12,  18 

and  78  ? 

12  =  ^  X  t?  X  ^  Solution.  —  By  factor- 

/;?  =  ^  X  "?  X  ^  ing  we  find  the  prime  fac- 

tors  common  to  12,  18  and 

78  — 2  X  3X  13  78  are  2  and  3. 

Greatest  Com.  Div.    ^^2X3  =  6  Since  the  product  of 

p.  all  the  common  prime  fac- 

'    io\i(0     1  o    lyo  ^^^^  ^^  *^''^  greatest  com- 

? 1 mon   divisor    (Art.    113), 

3)    6y      9,  39  2  X  3,  or  6,  must  be  that 

">,     3,  13  ^^^^^•'^'■- 

Or,  dividing  by  2,  we 
take  out  that  common  factor.  Dividing  the  resulting  quotients  by  3,  we  take 
out  that  common  factor  and  obtain  quotients  that  are  prime  to  each  other. 
Hence  2  and  3  are  all  the  factors  common  to  the  given  numbers,  and  their 
product,  2  X  3,  or  6,  must  be  the  greatest  common  divisor  required. 


FACTORING.  69 

2.  What  is  the  greatest  common  divisor  of  91  and  133  ? 

91)  1S3(  1  Solution. — Since  any  number  is  the  great- 

Q  -I  est  divisor  of  itself,  if  91  be  a  diviisor  of  133, 

it  must   be  the   greatest  common   divisor  of 

42)91(2  91  and  133.     We  find  that  it  is  not  a  divisor 

84-  of  133,  since  on  trial  42  remains. 

7  )  /^/^P  ^^  '^'^  ^®  ^  divisor  of  91,  it  is  also  a  divisor  of 

Z)  133,  which  is  once  91,  plus  42.  (Art.  113—2.)  It 

_T is  not  a  divisor  of  91,  since  on  trial  7  remains. 

If  7  is  a  divisor  of  42,  it  must  be  also  a  divisor  of  91,  which  is  twice 
42,  plus  7.     On  trial  it  is  found  to  be  a  divisor  of  42. 

Now,  7  is  the  greatest  divisor  of  7  and  42 ;  hence,  7  is  the  greatest 
common  divisor  of  42  and  91,  and  therefore  7  is  the  greatest  common 
divisor  of  91  and  133. 

3.  What  is  the  greatest  common  divisor  of  84  and  132? 

117.  Rules  for  finding  the  Greatest  Common  Divisor.— i.  Find 
the  prime  factors  common  to  the  given  numbers,  and 
the  product  of  those  factors  will  he  the  greatest  coin- 
?non  divisor  required.     Or, 

2.  Divide  the  greater'  number  by  the  less,  and  if 
there  be  a  remainder ,  divide  the  divisor  by  it,  and  so 
continue  to  divide  the  last  divisor  by  the  last  remain- 
der till  an  exact  divisor  is  found.  That  divisor  will 
he  the  greatest  convmon  divisor  of  the  two  numbers. 

3.  If  more  than  two  numbers  are  given,  find  first 
the  greatest  common  divisor  of  two  of  theiiv,  and 
then  the  greatest  common  divisor  of  that  divisor  and 
another  of  the  numbers,  and  so  on  till  all  the  numbers 
have  been  used.  The  last  common  d,ivisor  will  he  the 
greatest  common  divisor  of  all  the  numbers. 

Pn01iZ,EMS. 

What  is  the  greatest  common  divisor  of — 

1.  32,  48  and  80?       Ans.  16.  !  4.  308  and  630?         Ans.  14. 

2.  75  and  165?  |  5.  91  and  117? 

3.  72  and  168  ?  Ans.  24.  I  6.  21,  30,  39  and  81  ?     Ans.  3. 


60  FACTORING. 

7.  I  have  three  rooms,  the  first  of  which  is  16  feet  wide ;  the 
second,  20  feet;  and  the  third,  24  feet.  What  must  be  the 
width  of  carpeting  which  will  exactly  fit  each  room  ? 

Ans.  4  feet. 

8.  There  is  a  garden  84  feet  wide  and  1068  feet  long. 
What  must  be  the  length  of  the  longest  rails  that  will,  without 
cutting,  exactly  enclose  it  ?  Ans.  12  feet. 

9.  Find  the  greatest  common  divisor  of  99  bushels,  261 
bushels  and  504  bushels.  Ans.  9  bushels. 

10.  James  has  66  dollars,  Edward  has  77  dollars,  and 
Arthur  has  264  dollars.  If  they  should  purchase  fleur  at 
the  highest  price  per  barrel  that  would  allow  each  to  exactly 
use  his  money,  how  much  would  the  flour  cost  per  barrel  ? 

Ans.  11  dollars. 


SECTION    XI. 
MULTIPLES. 

118.— Ex.  1.  What  number  is  7  times  6  ?     5  times  6  ? 

2.  What  number  is  some  integral  number  of  times  6  ? 

3.  What  prime  factors  have  6  and  7  times  6  in  common  ?  6 
and  5  times  6  ? 

4.  What  prime  factors  have  6  and  42  in  common  ?  6  and 
30? 

5.  What  two  numbers  are  contained  in  6  an  exact  number  of 
times  ?     Of  what  two  numbers  is  6  an  exact  number  of  times  ? 

6.  Of  what  two  numbers  is  15  an  exact  number  of  times? 
Of  what  number  are  those  two  numbers  prime  factors? 

7.  What  numbers  from  5  to  30  contain  5  an  integral  num- 
ber of  times  ? 

8.  What  numbers  from  6  to  24  contain  botli  2  and  3  an  in- 
tegral number  of  times  ? 

9.  What  is  the  least  number  that  contains  both  2  and  3  an 
integral  number  of  times? 

10.  What  is  the  least  number  that  is  an  integral  number  of 
times  3  and  4?  What  are  the  prime  factors  of  3  and  4? 
Of  12? 


FA  CTOEING.  61 

11,  What  is  the  least  number  that  is  an  integral  number  of 
times  10  and  6?  What  are  the  prime  factors  of  10  and  6 
which  when  taken  the  least  number  of  times  will  form  those 
numbers  ?     What  are  the  prime  factors  of  30  ? 

DEFINITIONS. 

119.  A  Multiple  of  a  number  is  any  integral  number  of  times 
that  number. 

Thus,  10,  which  is  twice  5,  is  a  multiple  of  5. 

120.  A  Coiinnon  Multiple  of  two  or  more  numbers  is  any 
number  which  is  an  integral  number  of  times  each  of  them. 

Thus,  12,  which  is  3  times  4,  is  a  common  multiple  of  3  and  4. 

121o  The  Least  Commou  Multiple  of  two  or  more  numbers  is 
the  least  number  which  is  aii  integral  number  of  times  each  of 
them. 

Thus,  6  is  the  least  common  multiple  of  2  and  3. 

Only  similar  numbers  can  have  a  common  multiple,  since 
numbers  must  be  similar  to  be  factors  of  the  same  product. 

122.  Principles. — 1.  A  multiple  of  a  number  contains  all  the 
prime  factors  of  that  number. 

2,  A  common  muUiple  of  two  or  more  numbers  contains  all  the 
prime  factors  of  those  numbers. 

3.  The  least  common  nniltiple  of  two  or  more  numbers  is  the 
least  number  that  contains  all  the  prime  factors  of  those  numbers. 

C^SE    I. 

Common  Multiples. 

123.  Ex.  1. — Find  a  common  multiple  of  9  and  11. 

^  -f  Ky  Q QQ  Solution. — Since    a    common    multiple   of   the 

given   numbers  must  be  a  number  which  contains 
those  numbers  as  factors,  their  product,  or  99,  is  a  common  multiple. 

2.  Find  a  common  multiple  of  5,  7  and  9.  Ans.  315. 

124.  Rule  for  finding  a  Common  Multiple.— Multipli/  the  given 
niiTYibers  together,  and  the  product  will  be  a  common 
multiple  of  those  numbers. 


62  FACTORING. 

PROBLEMS. 

1.  What  is  a  common  multijjle  of  3,  5  and  6?      Ans.  90. 

2.  Find  a  common  multiple  of  11  and  13. 

3.  What  is  a  common  multiple  of  7,  10  and  25  ? 

Ans.  1750. 

C^?k.SE    II. 

Least  Common  Multiples. 

125.  Ex.  1. — Find  the  least  common  multiple  of  4,  12 
and  30. 

j^---  2  y^  2  Solution. — A    multiple  of   4    must 

-iq) ^V  $?y   "?  contain  its   prime   factors   2   and   2;   a 

'*'        ^  s/   <z)  s/   /r  multiple  of    12   must   contain   the   ad- 

oO  ^  ^  ?\  3  y^  O  ditional  pnme  factor  3 ;  and  a  multiple 

'5*X^X'?X'>=6'^       of  30  must  contain  the  additional  prime 

factor  5. 
2,  2,  3  and  5  are  all  the  prime  factors  of  the  numbers;  hence,  the 
product  of  these  factors,  or  60,  is  their  least  common  multiple. 

2.  Find  the  least  common  multiple  of  42,  49  and  70. 

7 )A2    A9    70  Solution.— By  division  we 

«  \~^ ^y — Tn  ^^^^   ''"^   ^^^"^  prime  factor  7, 

■^       >      / ,   -t  1/  common  to  the  given  numbers, 

3,      7,      5  and  liave  left  the  factors  6,  7 

and  10. 
7  y^  2  y^  3  y^  7  y^  5  ^  147 0  We  take  out  the  prime  fac- 

tor 2,  common  to  6  and  10,  and  have  left  the  factors  3,  7  and  5,  which 
have  no  factor  common  to  any  two  of  them. 

Hence,  7,  2,  3,  7  and  5  are  all  the  prime  factors  of  the  numbers,  and 
their  product,  1470,  is  the  least  common  multiple  required. 

3.  Find  the  least  common  multiple  of  8,  11  and  15. 

Ans.  1320. 

126.  Rules  for  finding  the  Least  Common  Multiple.—  1.  Find  the 
prime  ftvctora  of  tlie  glueii  nwDibers,  ami  the  product 
of  the  different  prime  factors,  eacJi  factor  being  ta^hen 
the  greatest  numher  of  times  it  occurs  in  any  of  the 
numbers,  will  be  the  least  comm^ou  multiple.     Or, 

2.  Place  the  given  numbers  in  a  horizontal  line; 


FACTORING.  63 

divide  hy  any  prime  ninnher  that  is  a  factor^  of  two 
or  iy%ore  of  them,  and  write  the  quotients  and  un- 
divided numbers  heloiv.  Divide  these,  if  possible,  in 
lihe  manner,  and  so  continue  until  the  quotients  and 
undivided  numbers  are  prime  to  each  other.  TJie 
product  of  the  divisors  and  the  numbers  remaining 
in  the  last  horizontal  line  will  be  the  least  common 
multiple. 

PROBLEMS, 

What  is  the  least  commou  multiple  of — 


1.  24,  15  and  16?      Ans.  240. 

2.  42  and  56  ?  Ans.  168. 

3.  9,  11  and  48? 


4.  13  and  29?  .4ns.  377. 

5.  18,  27  and  30  ? 

6.  60,  50  and  35?  Am.  2100. 


7.  Find  the  least  commou  multiple  of  40,  36,  32,  30,  28,  24, 
20  and  18. 

40,  36,  32,  30,  28,  24-,  20,  18         Solution. -Write  40  as 

a  factor  of  the  answer.    The 

4^->  X  5  X  ^  X  7  =  10080  largest  factor  common  to  40 

and  3G  is  4;  write  9,  the  remaining  factor  of  36,  as  a  factor  of  the 
answer. 

The  largest  factor  common  to  40  and  32  is  8 ;  write  4,  the  remaining 
factor  of  32,  as  a  factor  of  the  answer. 

The  largest  factor  common  to  40  and  30  is  10;  and  3,  the  remaining 
foctor  of  30,  is  found  in  9. 

The  largest  factor  common  to  40  and  28  is  4;  write  7,  the  remaining 
factor  of  28,  as  a  factor  of  the  answer. 

The  largest  factor  common  to  40  and  24  is  8 ;  and  3,  the  remaining 
factor  of  24,  is  found  in  9. 

Tlie  factors  of  20  are  found  in  40,  and  the  factors  of  18  in  36.  The 
continued  product  of  the  factors  of  the  answer  is  10080. 

8.  Find  the  least  common  multiple  of  8,  10,  11,  90  and  132. 

9.  What  is  the  least  common  multiple  of  7,  16,  21  and  28? 

10.  Find  the  smallest  number  that  will  exactly  contain  9, 
15,  18  and  20.  Ans.  180. 

11.  What  is  the  least  number  of  cents  with  which  you  may 
purchase  either  slates  at  18  cents  or  arithmetics  at  63  cents 
each?  Am.  126. 


64  FACTORING. 

12.  What  is  the  smallest  sum  of  money  for  which  I  can 
purchase  calves  at  17  dollars  each,  yearlings  at  34  dollars 
each,  or  cows  at  68  dollars  each  ?  Ans.  68  dollars. 

13.  What  is  the  least  number  of  acres  that  can  be  exactly 
divided  into  lots  of  12  acres,  15  acres  or  16  acres  each? 

Ans.  240. 


TEST    QUESTIONS. 

127. — 1.  What  is  a  Common  Divisor  of  two  or  more  numbers  ?  Why 
cannot  4  dollars  and  6  yards  have  a  common  divisor  ? 

What  is  the  greatest  common  divisor  of  two  or  more  numbers  ?  What 
numbers  have  no  common  divisor  ? 

2.  What  is  the  Principle  in  relation  to  the  greatest  common  divisor 
of  numbers?  Of  what  is  the  divisor  of  a  number  a  divisor?  Of  what 
is  a  common  divisor  of  two  or  more  numbers  also  a  divisor? 

3.  What  is  the  Rule  for  finding  the  common  divisor  of  two  or  more 
numbers  ?  For  finding  the  greatest  common  divisor  of  two  or  more 
numbers  ? 

4.  What  is  a  Multiple  of  a  number?  A  common  multiple  of  two 
or  more  numbers?   The  least  common  multiple  of  two  or  more  numbers? 

5.  What  is  the  Principle  in  relation  to  a  multiple  of  a  number?  In 
relation  to  a  multiple  of  two  or  more  numbers  ?  In  relation  to  the  least 
common  multiple  of  two  or  more  numbers? 

6.  What  is  the  Rule  for  finding  a  common  multiple?  For  finding 
the  least  common  multiple  of  two  or  more  numbers?  When  numbers 
are  prime  to  each  other,  how  is  their  least  common  multiple  found  ? 


SECTION   XII. 
FACTORS  r.W  DiriSIOJ^. 

128.  The  Value  of  a  quotient  in  division  depends  upon  the 
relative  values  of  dividend  and  divisor.     Hence, 

Any  change  in  the  factor. i  of  the  dividend  or  divisor  must  affect 
the  value  of  the  quotient. 

Thus,  24-- 6  =  4; 

and  (24X2) -^6  =  8,     or,  24 -- (6 -- 2)  =  8; 

also,  (24 -f- 2) --6  =2,     or,  24 -f- (6  X  2)  =  2. 


FACTORING.  66 

The  same  change  in  the  factors  of  both  dividend  and  divisor 
does  not  affect  the  value  of  the  quotient. 

Thus,  (24  --  2)  --  (6  4-  2)  =  4 ;  or,  (24  X  2)  -^  (6  X  2)  =  4. 

129.  General  Principles  of  Division. — 1.  Multiplying  the 
dividend,  or  dividing  the  divisor,  multiplies  the  quotient. 

2.  Dividing  the  dividend,  or  vndtiplying  the  divisor,  divides 
tJie  quotie)U. 

3.  Dividing  or  multijohjing  both  dividend  and  divisor  by  the 
same  number  does  not  change  the  quotient. 

C^SK    I. 

Division  by  Factors. 

130.  Ex.  1.  Divide  9702  by  21,  using  the  factors  of  21. 

Solution. — The  factors  of  21  are  3  and  7.  Since  21 
3)9702  times  a  number  is  7  times  3  times  the  number,  one  twenty- 
_,  s  „„g  .        first  of  a  number  must  be  one  seventh  of  one  third  of  that 

J-        number. 

462  One   third   of  9702,   the   dividend,  is   3234,   and   one 

seventh  of  3234  is  462. 

2.  Divide  4677  by  45,  using  factors. 

5)A677  Solution. — 45  is  equal  to  5  X  9- 

, -•  Dividing  4677  by  5,  we  have  935/t>cs, 

9 J    9oO,  ^  ones  =     ^        ^^^^  2  ones  as  a  remainder. 

103,  8  fives  =■  40            Dividing  by  9,  we  have  103  forty-fives, 

_         --,  .     ,            "T^n        and  8  fives  as  a  remainder. 

True  Remainder,      4-"  rt^^     ^    ^       4.-  1  •  ^     •  o 

Tlie  first  })artial  remainder  is  2  ones,  or 

2,  and  the  second  partial  remainder  8  fives,  or  40 ;  hence,  2  +  40,  or  42, 

is  the  whole  or  true  remainder,  and  103||  is  the  quotient  required. 

The  factors  5,  3  and  3  could  have  been  used  with  the  same  result. 

3.  Divide  825  by  86,  using  factors.  Ans.  22f|. 

131.  Rule  for  Dividing  by  Factors.— i^//^,rZ  any  convenient  set 
of  factors  of  the  rlivisor ;  divide  tlie  dividend  hy  one 
of  these  factors,  and  the  quotient  thus  obtained  by 
another,  and  so  on.  till  all  the  factor's  are  used. 

If  there  be  onr  or  more  reri%a.ind;ers,  multiply  each. 

by  the  divisors  preceding  the  one  that  produced  it, 

and  add  the  prod^ucts  and  the  remainder,  if  any ,  from 

the  first  division.   The  sum  ivill  be  the  true  remainder. 

6* 


66  FACTORING 

PnOBLEMS. 

Divide,  using  factors — 

1.  2954  by  14.  Ans.  211. 

2.  3728  by  28.  Ans.  133f 

3.  8316  by  27. 

4.  88763  by  32.  Ans.  2773f|. 


5.  47839  by  42.   Ans.  1139^^ 

6.  11630  by  81. 

7.  2520  by  105.  Aiis.  24. 

8.  196473  by  72.  Am.  2728f|. 


9.  Divide  94596  by  2300,  using  the  tactors  23  and  100. 

ogg  SoLTJTiox. — Dividing  by  100,  we  have 

23 \  00J94-o\  96 (  4-1^^      945  hundreds,  and  96  ones  as  a  remainder. 
92  Dividing  by  23,  we  have  41  twenty-three 

^f-  hundreds,  and   2  hundreds  as  a  remain- 

der. 
^^  2  hundreds  +  96  ones  =  296,  tlie  true 

^  remainder,  and  '^1-^^^%  is  the  quotient 

required. 
In  the  computation  we  denote,  for  brevity,  the  factoring  of  tlie  divisor 
by  cutting  off  the  two  ciphers  by  a  mark,  and  in  the  division  of  the  divi- 
dend by  100  we  set  off  the  remainder  by  the  same  mark. 

10.  Divide  782967  by  3700,  using  the  factors  37  and  100. 

A71S.  21imi- 

11.  Divide  46370  by  90,  using  the  foctors  9  and  10. 

12.  What  is  the  quotient  of  345600  divided  by  5000  ? 

Am.  69^%%. 

13.  Divide  16632  by  5148,  using  the  factors  11,  18  and  26. 

14.  If  700  barrels  of  apples  cost  2100  dollars,  how  much 
will  one  barrel  cost?  ^1??^.  3  dollars. 

15.  If  63  bushels  of  wheat  make  one  load,  how  many  full 
loads  can  be  made  from  1937  bushels,  and  how  many  bushels 
will  remain  ?  Ans.  30  full  loads,  and  47  bushels  over. 


CASK    II. 

Cancellation. 

132.  Cancellation  is  the  process  of  shortening  computations 
by  striking  out  equal  factors  from  the  dividend  and  divisor, 
and  using  only  the  remaining  factors. 


FA  CTORING.  G7 

133.— Ex.  1.  Divide  11  X  3  X  2  by  11  X  2. 

1                   1  Solution.  —  Indicate    the 

ti-  X  3  XS-     1X3X1  division  by  writing  the  divi- 

y—T^ —  ~ 1    sy  -I —  ~  ^  ''^"^  °^'^^'  ^^^^  divisor. 

TTXlt               1X1  DWiAc  botli  dividend  and 

^  1  divisor  by  the  factors  11  and 
2,  by  cancelling  those  common  factors  in  both,  whicli  does  not  change  tlio 
quotient.    (Art.  129—3.) 

When  a  factor  is  cancelled,  1  remains,  and  if  not  written  is  understood. 

2.  Divide  15  X  6  X  7  by  10  X  18. 

Solution. — Cancelling  the  factor  5,  com- 
mon to  both  dividend  and  divisor,  we  have 
.  c,  1        in  the  dividend  3  in  place  of  15,  and  in  the 
~         divisor  2  in  place  of  10.     We   next  cancel 
3X6,  or  18,  common  to  both  dividend  and 
divisor,  and  have  left  -|,  or  Z\. 

3.  Divide  11  X  9  X  8  by  11  X  3  X  2.  Ans.  12. 

134.  Rule  for  Cancellation.— C«-?^ceZ  Uv  the  dividend  and 
divisor  all  factors  common  to  both,  and  then  divide  the 
product  of  the  remaining  factors  of  the  dividend  by 
the  product  pf  the  remaining  factors  of  the  diviso?\ 


PHOB  T.EMS. 

1.  Divide  35  X  8  X  3  by  12  X  7  X  5. 

2.  Divide  42  X  15  X  80  by  75  X  24.  Ans.  28. 

3.  Divide  96  X  63  X  5  by  72  X  35. 


4.  Di^dde  108  X  77  X  2  by  18  X  26  X  H.        Ans. 


'J 1  :r 


5.  Divide  65  X  16  X  33  by  26  X  22  X  8. 

6.  How  many  are  (300  X  45  X  6)  --  (150  X  30  X  18)  ? 

7.  How  many  barrels  of  beef,  at  21  dollars  a  barrel,  are 
worth  as  much  as  14  tons  of  coal,  at  6  dollars  a  ton  ?    Ans.  4. 

8.  How  many  bushels  of  corn,  at  90  cents  a  bushel,  will  pay 
ibr  120  yards  of  cloth,  at  15  cents  a  yard  ? 

9.  I  exchanged  90  bushels  of  potatoes,  at  75  cents  a  bushel, 
for  tubs  of  butter  containing  54  pounds  each,  at  25  cents  a 
pound.     How  many  tubs  of  butter  did  I  receive  ?      Ans.  5. 


68  FA  CTORING. 

SECTION    XIII. 

AXALYSIS. 

135. — Ex.  1.  If  24  bushels  of  wheat  cost  72  dollars,  what 
v.ill  5  bushels  cost? 

Solution. — If  24  bushels  cost  72  dollars,  1  bushel  will  cost  one 
twenty-fourth  of  72  dollars,  which  is  3  dollars.  If  1  bushel  cost  3 
dollars,  5  bushels  will  cost  5  times  3  dollars,  which  are  lo  dollars. 

2.  If  15  men  can  earn  75  dollars  in  a  given  time,  how  much 
can  7  men  earn  in  the  same  time  ? 

3.  When  7  hats  cost  35  dollars,  how  much  will  15  hats  cost? 

4.  If  7  hats  cost  35  dollars,  how  many  hats  will  cost  75 
dollars  ? 

Solution. — If  7  hats  cost  35  dollars,  1  hat  will  cost  \  of  35  dollars, 
which  is  5  dollars,  and  as  many  hats  will  cost  75  dollars  as  5  dollars  are 
contained  times  in  75  dollars,  which  are  15. 

5.  If  4  persons  require  48  dollars'  worth  of  provisions  in  a 
certain  time,  how  many  persons  Avill  require  108  dollars'  worth 
in  the  same  time  ? 

6.  When  13  bushels  of  corn  are  worth  as  much  as  39 
bushels  of  oats,  how  many  bushels  of  corn  are  worth  as  much 
as  27  bushels  of  oats  ? 

7.  If  17  men  can  earn  as  much  in  one  day  as  51  boys,  who 
are  each  paid  50  cents  a  day,  how  much  can  1  man  earn  in  a 
day? 

8.  If  9  barrels  of  flour  are  worth  as  much  as  36  cords  of 
wood,  how  many  cords  of  wood  are  worth  as  much  as  7  barrels 
of  flour? 

9.  When  28  cords  of  wood  are  worth  as  much  as  7  barrels 
of  flour,  how  many  cords  of  wood  are  worth  as  much  as  0 
barrels  of  flour  ? 

DEFINITION. 

136.  Analysis,  in  Arithmetic,  i.s  the  process  of  stating,  in 
regular  order,  the  reasons  for  each  step  in  the  solution  of  a 
problem  whose  conditions  require  several  computations. 


FA  CTOBING.  69 

WRITTEN   JSXIDRCISES. 

Ex.  1.  If  12  horses  cost  2100  dollars,  liow  much  will  21 
horses  cost  at  the  same  rate  ? 

12)2100  dollars.  Soi^ution  by  Anai^ysis.-K  12 

— - — -  horses  cost  2100  dollar.s,  1  horse  will 

-^  '  ^  cost  one  twelfth  of  2100  dollars,  which 

21  is  175  dollars. 


1  ly  n  If  1  horse  cost  175  dollars,  21  horses 

^r-^  will  cost  21  times  175  dollars,  whicli 

are  3675  dollars. 

367 5  dollars.  Solution  by  CANCf:Li,ATioN. — If 

Or,  12  horses  cost  2100  dollars,  1  horse 

525  7  will  cost  one  twelfth  of  2100   dol- 

jnirin-y  ~)-j-  lars,   and    21    horses    will    cost    21 

=^S075       times  as  much.    Indicating  the  work, 

^^  and   cancelling,  we   have  3675,  the 

^  same  result. 

2.  When  45  tons  of  coal  cost  270  dollars,  what  will  60  tons 
cost?  Ans.  360  dollars. 

3.  If  180  quarts  of  oats  be  sufficient  for  30  horses  for  a  cer- 
tain time,  how  many  quarts  will  be  sufficient  for  63  horses  for 
the  same  time  ? 

4.  If  15  men  can  do  a  piece  of  work  in  54  days,  in  what 
time  can  9  men  do  the  same  ? 

5.  What  time  will  48  men  require  to  mow  a  field  that  10 
men  can  mow  in  32  days  ?  Ajas.  6f  days. 

6.  A  cistern  can  be  filled  in  265  minutes  by  5  equal  pipes 
running  into  it.  In  what  time  could  it  be  filled  by  25  such 
pipes?  Ans.  53^ minutes.     . 


TEST    QUESTIONS. 

137. — 1.  Upon  what  does  the  Value  of  a  quotient  depend?  What 
effect  upon  the  quotient  is  produced  hy  multiplying  the  dividend  or 
dividing  the  divisor  ?  What  effect  by  dividing  the  dividend  or  multiply- 
ing the  divisor?  What  changes  in  both  dividend  and  divisor  do  not  affect 
the  value  of  the  quotient  ? 

2.  How  do  you  divide  by  using  Factors?  If  there  be  one  or  more 
remainders,  liow  do  you  find  the  true  remainder  ? 

3.  What  is  Cancellation  ?    The  rule  for  cancellation  ?     Analysis  ? 


70  REVIEW    PROBLEMS. 

-   SECTION    XIV. 
REVIEW  PROBLEMS. 

MENTAL  EXERCISES. 

138. — Ex.  1.  Of  what  numbers  are  2,  3  and  11  the  prime 
factors  ? 

2.  What  are  all  the  exact  divisors  of  66  ? 

3.  Name  the  composite  numbers  from  2  to  30.  From  30 
to  45. 

4.  Five  20's  are  how  many  4's  ?  Seven  12's  are  how  many 
14's? 

5.  I  sold  9  oranges  at  8  cents  apiece,  and  spent  the  money 
for  melons  at  18  cents  each.  Hoav  many  melons  did  I  pur- 
chase ? 

6.  What  common  factors  have  18  and  42? 

7.  What  is  the  greatest  number  that  will  exactly  divide  27 
and  63  ? 

8.  When  cheese  is  15  cents  a  pound,  and  butter  35  cents, 
what  is  the  least  number  of  pounds  of  cheese  that  can  be  ex- 
changed for  an  exact  number  of  pounds  of  butter  ? 

9.  When  bvitter  is  45  cents  a  pound,  how  many  pounds  of 
sugar,  at  18  cents  a  pound,  will  cost  as  much  as  2  pounds  of 
butter  ?  '^ 

10.  If  5  men  can  perform  a  piece  of  work  in  14  days,  in 
what  time  can  7  men  do  it  ? 

11.  If  8  men  can  do  a  piece  of  work  in  15  days,  how  many 
men  can  do  it  in  12  days  ? 

12.  When  coffee  is  35  cents  a  pound,  how  many  pounds  of 
sugar,  at  14  cents  a  pound,  will  cost  as  much  as  2  pounds  of 
coffee  ? 

13.  If  6  men  can  do  a  piece  of  work  in  14  days,  how  many 
men  can  do  it  in  7  days  ? 

14.  I  sold  11  pears  at  6  cents  each,  and  bought  with  the 
money,  oranges  at  3  cents  each.  How  many  oranges  did  I 
buy? 


-REVIEW    PROBLEMS.  71 

WRITTEN   EXERCISES. 

139.— Ex.  1.  What  are  the  prime  factors  of  56? 

2.  Which  of  the  numbers  31, 98  and  101  are  prime  numbers  ? 

3.  Find  the  greatest  common  divisor  of  4165  and  686. 

4.  Which  of  the  numbers  700,  575  and  335  have  25  for  an 
exact  divisor  ? 

5.  Find  all  the  exact  divisors  of  90. 

6.  What  is  the  least  common  multiple  of  8,  11  and  13? 

Ans.  1144. 

7.  What  exponent  will  denote  the  number  of  times  that 
5  is  taken  as  a  factor  in  3125  ? 

8.  I  have  14  bushels  of  oats,  22  bushels  of  rye  and  24 
bushels  of  corn.  What  is  the  capacity  of  the  largest  sacks,  of 
equal  size,  into  which  the  whole  may  be  put  without  mixing  ? 

Ans.  2  bushels. 

9.  Three  men  start  at  the  same  time  and  place  to  walk  in 
the  same  direction  round  a  circle.  A  can  make  the  circuit 
in  5  hours,  B  in  8  hours,  and  C  in  10  hours.  In  what  time 
after  they  start  will  they  all  be  together  again  at  the  point  of 
starting?  Ans.  40  hours. 

10.  If  1350  men  can  build  a  bridge  in  30  days,  in  what 
time  can  1500  men  build  it? 

11.  When  the  cost  of  42  pounds  of  rice  is  294  cents,  what 
is  the  cost  of  28  pounds  ?  Ans.  196  cents 

12.  How  many  tons  of  ii'on  can  be  bought  for  95285  dollars, 
if  50  tons  can  be  bought  for  4250  dollars?  Ans.  1121. 

13.  If  27  men  can  earn  2187  dollars  in  one  month,  how 
many  men  can  earn  5832  dollars  in  the  same  time  ? 

Ans.  Tl. 

14.  If  18  men  can  sow  a  field  of  oats  in  12  days,  how  long 
will  it  take  48  men  to  sow  a  field  of  the  same  size  ? 

15.  How  many  pounds  of  tea,  at  80  cents  a  pound,  are 
equal  in  value  to  6  bushels  of  wheat,  Avorth  280  cents  a  bushel  ? 

16.  I  exchanged  15  pieces  of  cloth,  each  containing  30  yards 
worth  14  cents  per  yard,  for  21  barrels  of  apples,  each  con- 
taining 3  bushels.  How  much  did  the  apples  cost  me  per 
bushel  ?  Ans.  100  cents. 


72 


FRACTIONS. 


SECTION    XV. 

COMMO.Y  FR.  i  CTIOjYS. 

140. — Ex.  1.  If  an  apple  be  cut  into  two  equal  parts,  what 
part  of  the  apple  will  one  of  the  pieces  be  ? 

2.  What  is  one  half  of  an  apple,  or  one  half  of  anything  ? 
One  third  of  an  apple,  or  one  third  of  anything  ? 

3.  One  of  the  three  equal  parts  of  an  apple  is  what  part  of 
the  whole  ?  Two  of  the  three  equal  parts  are  what  part  of  the 
whole  ? 

4.  What  is  one  fourth  of  anything?  Two  fourths?  Three 
fourths  ? 

5.  How  many  halves  in  a  single  thing  or  unit?  How  many 
thirds  ?     How  many  fourths  ? 

6.  What  is  one  fifth  of  a  unit?  One  sixth  ?  One  seventh? 
One  eighth  ? 

7.  What  are  two  fifths  of  anything?  Three  fiftlis?  Four 
fifths? 

8.  Which  is  the  greater — a  half  or  a  third  ?  A  third  or  r^ 
fourth  ?     A  third  or  a  fiftli  ? 

9.  How  are  halves  expressed  by  figures?  Thirds?  Fourths? 
Fifths? 

10.  What  does  "I"  signify  ?     What  does  f  signify  ? 


FRACTIONS.  73 

11.  In  f ,  what  expresses  the  number  of  parts  into  which  a 
unit  has  been  divided  ?  What  expresses  the  number  of  parts 
taken  ? 

DEFINITIONS. 

141.  A  Fraction  is  a  number  which  expresses  one  or  more 
of  the  equal  parts  into  which  a  unit  is  divided. 

Thus,  one  half,  two  thirds,  five  fourths,  etc.,  are  fractions. 

142.  The  Unit  of  the  Fraction  is  'the  unit,  or  whole  thing, 
which  is  considered  as  di^♦ded  into  parts. 

Thus,  tlie  unit  of  tiie  fraction  of  an  apple  is  one  apple ;  the  unit  of 
the  fraction  of  a  dollar  is  one  dollar,  etc. 

When  no  particular  unit  is  named,  the  abstract  unit  1  is 
understood. 
Thus,  halves,  thirds,  etc.,  are  understood  to  be  halves,  thirds,  etc.,  of  1. 

143.  A  Fractional  Unit  is  one  of  the  equal  parts  of  the  unit 
of  the  fraction. 

Thus,  one  half,  one  third,  etc.,  is  the  fractional  unit  of  halves,  thirds,  etc. 

144.  The  Penominator  of  a  fraction  is  the  number  which 
denominates  or  names  the  parts  of  the  unit. 

Thus,  four  is  the  denominator  of  f . 

145.  The  T^mnerator  of  a  fraction  is  the  number  which 
numerates  or  numbers  the  fractional  units  taken. 

Thus,  three  is  the  numerator  of  |. 

14(>.  The  Terms  of  a  fraction  are  its  numerator  and  denom- 
inator. 

147.  A  CoKimwi  Fraction  is  an  expression  of  any  number  of 
parts  of  a  unit,  written  by  placing  the  numerator  above  the 
denomin*ator,  with  a  line  between  them. 

Thus,  f,  wliich  is  an  expression  of  two  fifths,  is  a  common  fraction. 

148.  A  Proper  Fraction  is  one  whose  numerator  is  less  than 
its  denominator ;  and  an  Improper  Fraction  is  one  whose  nume- 
rator is  not  less  than  its  denominator. 

Thus,  f,  f,  etc.,  are  proper  fractions,  and  f,  f,  etc.,  are  improper  frac- 
tions. 


74  FRA  CTIONS. 

119.  A  Mixed  Number  is  a  number  expressed  by  an  integer 
and  a  fraction  ;  as,  5f . 

150.  Similar  Fractions  are  such  as  have  the  same  denomina- 
tor ;  as,  I,  I,  etc. 

151.  Dissimilar  Fractions  are  such  as  have  different  denomi- 
nators ;  as,  f ,  |,  etc. 

152.  An  Integer  may  be  expressed  fractionally  by  writing  it 
as  a  numerator,  with  1  as  its  denominator. 

Thus,  3  may  be  expressed  f,  and  be  rea^three  ones,  or  three. 

153.  Fractions  are  read  by  pronouncing  the  number  in  the 
numerator,  and  then  naming  the  parts  denoted  by  the  denom- 
inator. 

Thus,  I  is  read  one  half;  f,  tivo  thirds  ;  ^j,  three  twenty-firsts,  etc. 
151.  Fractions  may  not  only  bo  regarded  as  a  number  of 
parts  of  a  unit,  but  as  another  method  of  indicating  division. 
Thus,  4  may  be  regarded  as  f  of  1 ;  -|-  of  3 ;  or  3  divided  by  5. 

155.  Principles. — 1.  The  value  of  a  fraction  is  the  quotient 
obtained  hij  dividing  the  numerator  by  the  denominator. 

2.  The  value  of  a  fraction  is  less  than  1  rvhen  the  numerator 
is  less  than  the  denominator. 

3.  The  value  of  a  fraction  equals  or  exceeds  1  ivhen  the  nume- 
rator equals  or  exceeds  the  denominator. 

EXERCISES. 

156.  Name  the  kind  of  fraction,  and  read — 
1. 


s 

4. 

ft 
•J* 

7.  1^. 

10.  ||. 

5. 

1 .? 

T4- 

8.  i^. 

11.  ^. 

^. 

6. 

ToTF- 

9.  19t. 

12.  36^ 

3. 

Write  in  figures — 

13.  Nine  tenths. 

14.  Seven  eighths. 

15.  Eleven  thirteenths 

19.  Six  thousand  one  nineteenths. 

20.  How  may  3  be  expressed  fractionally? 

21.  Is  the  value  of  ^  greater  or  less  than  1  ?    Of  j%? 


16.  21  seventy-seconds. 

17.  Three  one-hundredths. 
IS.  1 7  t wo-hundrcd-twentieths. 


FRACTIONS.  76 

SECTION   XVI. 
REDUCTIOJV  OF  FRACTIOKS. 

157,  Reduction  of  Fractions  is  the  process  of  changing  their 
form  of  expression  without  changing  their  value. 

C^SE    I, 

Fractions  Reduced  to  Larger  or  Smaller  Terms. 

158. — Ex.  1.  One  third  of  an  apple  is  how  many  sixths  of 
an  apple  ? 

2.  What  is  \  expressed  in  terms  twice  as  large  ? 

3.  Express  \  in  terms  twice  as  large,  f  in  terms  3  times  as 
large. 

4.  Two  sixths  of  an  apple  are  how  many  thirds  of  an  apple? 
What  is  "I  expressed  in  terms  one  half  as  large  ? 

5.  Two  eighths  are  how  many  fourths?  Express  -^  in 
terms  one  third  as  large. 

6.  Ten  fifteenths  are  how  many  fifths  ?  Express  -^  in  terms 
one  sixth  as  large. 

DEFINITIONS. 

159.  A  fraction  is  reduced  to  Larger,  or  Higher,  Terms  when 
expressed  in  an  equivalent  fraction  with  larger  terms. 

160.  A  fraction  is  reduced  to  Smaller,  or  Lower,  Terms  when 
expressed  in  an  equivalent  fraction  with  smaller  terms. 

161.  A  fraction  is  in  its  Lowest  Terms  when  expressed  in 
terms  which  are  prime  to  each  other. 

162.  Principle. — Multiplying  or  dividing  both  terms  of  a 
fraction  by  the  same  number  does  not  change  its  value. 

For,  in  the  one  case,  as  the  number  of  parts  is  increased,  their  size 
is  diminished ;  and,  in  the  other  case,  as  the  number  of  parts  is  dimin-* 
ished,  their  size  is  increased. 


Thus    ■^=^.  -^^^  =  -^-<9  6^3  _2_^ 


76  FRA  CTIONS. 

WJtITTEN   EXEJtCTSES. 

163. — Ex.  1.  Reduce  f  to  forty-seconds. 

4  X  g 24_  Solution. — Since  42,  the  required  denominator,  is  6 

7X5  42  times  as  large  as  7,  the  given  denominator,  and,  since 
multiplying  both  terms  of  a  fraction  by  the  same  immber  does  not 
cliange  its  value,  (Art.  162,)  we  multiply  both  terms  of  ^  by  6,  which 
gives  If,  the  fraction  required. 

2.  Change  f  to  an  equivalent  fraction  whose  numerator  is  20. 

3.  Reduce  |4  to  its  lowest  terms. 

24^2       12     12^3  _  4  Solution.— Since  dividing  both  terms  of 

43-^2  21'  21-^3  7  a  fraction  by  the  same  number  does  not 
change  the  value  of  the  fraction,  we  divide  both  terms  of  |f  by  2,  or 
cancel  that  factor  in  each  term,  which  gives,  as  the  fraction  in  lower  terms, 
\\.  Dividing  both  terms  of  if  by  3  gives  4,  which,  since  the  terms  are 
prime  to  each  other,  is  the  result  required. 

24^6 4         Second   Solution. — Since  6  is  the  greatest  common 

42-^6      7      divisor   of  the  terms  of  ff,  we  can   obtain   the  lowest 

terms  of  the  fraction  by  dividing  both  terms  by  that  divisor,  which  gives 

f,  the  same  result  as  at  first  obtained. 

4.  Reduce  |^  to  its  lowest  terms. 

5.  Change  ff  to  its  lowest  terms. 

164;.  Rules  for  Reduction  of  Fractions  to  Higher  or  the  Lowest 
Terms.— i.  To  reduce  a  fraction  to  higher  temns,  multi- 
ply both  terms  of  the  fraction  hy  such  a  number  as 
will  give  the  required  term. 

2.  To  reduce  a  fraction  to  its  lowest  terms,  cancel  in 
both  terms  all  common  factors,  or  divide  both  terms 
by  their  greatest  common  divisor. 

rnOBLEJTS. 

1.  Reduce  ^  to  forty-ninths.  Ans.  ff. 

2.  Reduce  f  and  ^  to  sixteenths.  A)is.  y^,  \^. 

3.  Reduce  f  and  |  to  twenty-seventlis. 

4.  Reduce  ^,  ^  and  .^-f  to  one-hundred-fifths. 


FEA  CTIONS.  77 

Reduce  to  lowest  terms — 


5.  11^.  .      Ans.  f. 


TS- 


8.  ^.  Am.  If. 

9.  Hi.  Ans.  If. 

10  1702 

^^-  188G- 


7      ^82  J „ „      13 

11.  In  what  lower  terms  can  ^  be  expressed? 


J„,      1  2       8         6         4     orirl    2 

C^SE   II. 
Integers  or  Mixed  Numbers  Reduced  to  luiproper  Fractions. 

165. — Ex.  1.  How  many  fourths  in  1  orange?    In  3  oranges  ? 
In  6  oranges  ? 

2.  How   many  thirds  in   1   apple  ?      In  4  apples  ?      In   7 
apples  ? 

3.  How  many  fifths  in  1  ?     In  3  ?     In  6  ? 

4.  How  many  sixths  of  a  cake  are  2  cakes  ?     Are  2^  cakes  ? 
3-|-  cakes  ?     3f  cakes  ? 

5.  How  many  sevenths  in  1  ?     In  1\1     In  5f  ? 

WRITTEN  EXERCISES. 

166. — Ex.  1.  Reduce  29  to  sixths. 

29 

6  sixths  Solution. — Since  in  1  there  are  6  sixths,  in 

■fn,,   o/r//)?  =  ^~-^  29  there  must  be  29  times  6  sixths,  which  are  J-|^. 

2.  Reduce  18f  to  an  equivalent  improper  fraction. 

r  r 

18~  =  18-\ — ;  Solution. — Since  in  1  there  are  |,  in 

18  there  must  be  18  times  f,  or  J-f^ ;  i|^ 
XS=—;  ^  -f  ^  =  ^       and  f  are  ^\  the  fraction  required. 

3.  Reduce  19  to  fifths.  Ans.  ¥ 


4.  Change  41f  to  an  improper  fraction.  A 


ns. 


33  1 


167.  Rule  for  Reduction  of  Integers  or  Mixed  Numbers  to  Improper 
Fractions.— MiUfiply  the  integer  hy  the  given  denomi- 
nator, and,  if  there  he  a  fractional  part,  ad,d  its  nu- 
merator to  the  product.  Tlie  result  ivritten  over  the 
given  denominator  will  he  the  required  frjoiction. 
7* 


78  FRACTIONS. 

PROBLEMS. 

Reduce  to  equivalent  improper  fractions — 


1.  104f.      Am.  ^^. 

2.  28f 

3.  19i§i-.      ^*^-  W- 


5.  42f. 

6.  23^.      Ans.  ^-H- 

7.  28H- 


4.  1444.  8-  115tI?-   ^^• 


14  3  7  8 


125 


9.  Change  36  to  thirteenths  and  41  to  fourteenths. 

A  n't    4_68     5J4 
CASE  III. 

Improper  Fractions  Reduced  to  Integers  or  Mixed  Numbers. 
lg8,_Ex.  1.   In  four    fourths    of  an    orange,  how   many 
oranges  ?    In  twelve  fourths  of  an  orange  ? 

2.  How  many  apples  in  three  thirds  of  an  apple  ?   In  twelve 
thirds  of  an  apple? 

3.  How  many  ones  in  f  ?     In  f  ?     In  |  ?     In  ^2  ?    In  \^  ? 

4.  How  many  cakes  in  i/  of  a  cake  ?     In  ^3  of  a  cake  ? 

WRITTEN  EXERCISES. 

169. — Ex.  1.  Reduce  J-f^  to  an  e(iuivalent  integer  or  mixed 
number. 

~  :=  XQ'y  -^  g  ^  18~  Solution.— Since  9  ninths  equal  one,  167 

^  ^        ninths  must  equal   as  many  ones  as  9  is 

contained  times  in  167,  or  18|  times.     Hence,  ip  ^^  18|. 

2.  Reduce  ^/  to  an  equivalent  integer  or  mixed  number. 

170.  Rule  for  Reduction  of  Improper  Fractions  to  integers  or  Mixed 
Yi\xm\s^vs.— Divide  the  numerator  by  the  djenoirhinv^tor . 

PROBLEMS. 

Reduce  to  an  integer  or  mixed  number — 


1.  ^|i.  Ans.  129. 

3.  V^.  Am.  2|i 


4.  ifi.  Am.  151. 

5.  H^. 

6.  "tV-  ^'i«-  28|i. 


7.  What  is  the  value  of  ^^  dollars?        Am.  31  doHan 

8.  How  many  miles  are  ^^^  miles  ? 


FEA  CTIONS.  79 

C^SE    IV, 

Dissimilar  Fractions  Reduced  to  Similar  Fractions. 

171. — Ex.  1.  How  many  sixths  of  an  apple  is  1  third  of  au 
apple  ?     Are  2  thirds  of  an  apple  ? 

2.  Express  f  and  ^  each  as  sixths. 

3.  Express  \  and  f  each  as  sixteenths. 

4.  What  is  a  common  multiple  of  the  denominators  of  \ 
and  f  ? 

5.  Express  \,  f  and  f  each  as  twelfths.  What  is  tlie  least 
common  multiple  of  the  denominators  of  ^,  f  and  ^  ? 

DEFINITIONS. 

172.  Fractions  have  a  Common  Denominator  when  their  de- 
nominators are  alike. 

173.  Fractions  have  the  Least  Common  Denominator  when 
their  denominators  are  the  smallest  that  they  can  have  in 
common. 

174.  Fractions  are  said  to  be  reduced  to  a  common  denomi- 
nator when  they  are  changed  to  equivalent  fractions  with 
denominators  alike. 

175.  Principles, — 1.  A  common  denominator  of  hvo  or  more 
fractions  is  a  covimon.  multiple  of  their  denominators. 

2.  The  least  common  denominator  of  two  or  more  fractions  is 
the  least  common  multiple  of  their  denominators. 

WJRITTJEN  EXERCISES. 

176. — Ex.  1.  Reduce  f  and  ^  to  equivalent  fractions  having 
a  common  denominator. 

5^^  15  Solution.— Since  the  denominator  of  27  i'^  '^  times  tlie 

^       ^7  denominator  of  f,  we  multiply  both  terms  of  ~  by  3,  which 

J_^  gives  as  its  equivalent  \^.     Hence,  W  and  2:V  are  the  fractions 

'  required. 

2.  Reduce  ^4*,  |-  and  -^  to  similar  fractions. 

Ans.  f|,  If  and  ^. 


80  FHA  CTIONS. 

3.  Reduce  f  and  i  to  similar  fractions. 

^ ^  Solution. — Multiplying  both  terms  of  J  by  o,  the  denoni- 

4       ~0  inator  of  \,  we  have  if ;  and  multiplying  both  terms  of  i  by 

Jl  __  _i_  4,  the  denominator  of  |,  we  have  v.'V-     Hence,  |  and  ^  ^  ^| 

^       ^^  and  2*jy,  which  are  similar  fractions. 

4.  Reduce  -|^,  f  and  \^  to  equivalent  fractions  having  the 
least  common  denominator. 


iy.12  _  13_ 
2x12'    24 

5X3        15 


Solution. — The  least  common  multiple  of  the  de- 
nommators  2,  8  and  12  is  24;  hence,  24  is  the  least 
common  denominator. 

\  reduced  to  twenty-fourths  is  || ;  f  is  \\ ;  and  {\  is 


8XS        24 

11X2       22 

ja^a  ^~2l       ^1  >  lience,  if,  \\  and  ||  are  the  fractions  required. 

5.  Reduce  yV,  xi  ^^^^  It  *°  ^^®  least  common  denominator. 

J  ,,  c.        5  0         8  8         .5  5 

177.  Rules  for  Reducing  Fractions  to  a  Common  Denominator. — 

1.  Multiply  hotlh  terms  of  oue  or  more  of  the  frac- 
tions by  any  muinher  that  will  malce  the  denomiivators 
alike.    Or, 

2.  Multiply  both  temns  of  each  fraction  by  the  de- 
nomiivators  of  the  other  fractions. 

178.  Rule  for  Reducing  Fractions  to  the  Least  Common  Denom- 
inator.— Find,  the  least  common  multiple  of  all  the 
denominators  for  the  least  coi^vmoii  denominator,  and 
multiply  both  terms  of  each  fraction  by  such  a  num- 
ber as  ivill  reduce  it  to  that  denominatoj\ 

FJtOBT^EMS. 

Reduce  to  equivalent  fractions  having  a  common  denomina- 
tor— 

1       1    i    1  ^„<j    JU)       270      12fi. 

2-  h  h  i- 

3.  -I  and  :^.   ^  ^    ^  Am.  -}|,  ^V 

4.  Reduce  f ,  {^  and  y\  to  similar  fractions. 


Reduce 


TS" 


FRACTIONS.  81 

Reduce  to  similar  fractions  having  the  least  common  denom- 
inator— 

Art<i     27      20      21        8 
.a.11^.  -j^,    3g,  ^g,    3g. 


6. 

3     5 
¥'    9' 

f  2  and  tV- 

7. 

h  h 

f  and  |. 

8. 

5     3 
■g"'   8"' 

i  and  i|. 

q 

3     2 

¥'  1t2' 

o. 

T'  -g-' 

/1^,<f     216     126       56       966 
Ji^ns.  -g^-jy^,  -g-Q-j,  -5^:f,  3"^^. 

10.  Exjjress  7^,  5^  and  1^  as  similar  fractions,  with  the  least 
common  denominator.  Ans.  |f ,  ff ,  \^. 

TEST    QUESTIONS. 

1/9. — 1.  How  does  a  Fraction  differ  from  an  integer?  How  does 
the  unit  of  a  fraction  differ  from  a  fractional  unit?  What  is  the  unit 
and  what  is  the  fractional  unit  of  |  ? 

2.  What  are  the  Terms  of  a  fraction  ?  Which  term  names  the  parts  ? 
Which  term  numbers  the  parts  ?  What  is  the  numerator  and  what  is 
the  denominator  of  |  ? 

3.  How  is  a  Common  Fraction  written  ?  When  is  a  fraction  called 
a  proper  fraction  ?  When  an  improper  fraction  ?  What  is  a  mixed 
number  ? 

4.  How  is  a  common  fraction  expressed  ?  How  may  an  integer  be 
expressed  fractionally  ?  What  are  similar  fractions  ?  Dissimilar  frac- 
tions ? 

5.  How  is  a  fraction  read  ?     In  what  three  ways  may  -|  be  regarded  ? 

6.  Upon  what  does  the  Value  of  one  of  the  parts  of  a  fraction  de- 
pend ?  What  is  the  value  of  a  fraction  ?  When  is  the  value  of  a  frac- 
tion 1  ?     When  less  than  1  ?     When  more  than  1  ? 

7.  What  is  Reduction  of  fractions?  When  is  a  fraction  reduced  to 
higher  terms  ?     When  to  lower  terms  ? 

8.  When  is  a  fiaction  in  its  Lowest  Terms ?  How  may  a  fraction  be 
reduced  to  higher  terms  ?  How  to  its  lowest  terms  ?  Upon  what  prin- 
ciple does  the  rule  for  tlie  reduction  of  fractions  to  higher  or  to  lower 
terms  depend  ? 

9.  How  is  an  Integer  or  Mixed  Number  reduced  to  an  improper 
fraction  ?  How  is  an  improper  fraction  reduced  to  an  integer  or  mixed 
number? 

10.  When  have  several  fractions  a  Common  Denominator  ?  When 
the  least  common  denominator  ?  What  is  the  least  common  denomina- 
tor of  several  fractions  ?  What  is  the  rule  for  finding  common  denomi- 
nators? 


82  FRACTIONS. 

SECTION  XVIL 
ADDITIOJY  OF  FBACTIOJ^S. 

180. — Ex.  1,  James  gave  his  brother  3  eighths  of  a  melon 
and  his  sister  4  eighths.  What  part  of  the  melon  did  he  give 
both  ? 

2.  John  had  |-  of  a  dollar,  and  his  father  gave  him  f  of  a 
dollar.     What  part  of  a  dollar  had  he  then  ? 

3.  How  many  eighths  are  f  and  f  ?  How  many  tenths  are 
I  and  I  ? 

4  What  is  the  sum  of  3^  and  2|  ?    Of  |  +  i  +  2  +  3  ? 

181.  Principle. —  Only  fractions  having  a  common  denomi- 
nator can  be  directly  added. 

For,  fractions  having  a  common  denominator  are  similar,  and  only 
similar  numbers  can  be  added.  (Art.  46 — 1.) 

WRITTEN  EXERCISES. 

182.— Ex.  1.  What  is  the  sum  of  i  f  and  |? 

1  j_  4  ,    S  _  20  .    32       15  Solution.  —  Keducing    the    given 

2  5       8       40       40       40       fractions  to  their  least  common  denom- 
_  20  +  32  +  15  __67__  -.27  inator,  we    have    f§  +  ff  +  ^§  =  f^, 

40  ~~  40  ^     40  or  IfJ,  which  is  the  sum  required. 

2.  What  is  the  sum  of  ^,  -^  and  {^  ?  Am.  1^. 

3.  What  is  the  sum  of  3f ,  f  and  15  ? 

8            24  Solution. — Reducing  the  fractions  to  similar  frac- 

^  ^        i^  tions,  we  have  3|  =  3if ,  and  f  =  ^f . 

*             ^4  The  sum  of  these  fractions  is  f|^,  which  equals  l^^; 

15  =  15  and  1  ^^  added  to  18,  the  sum  of  the  integers,  gives  lO^'j, 

c,  y       the  sum  required. 

Sam,  19J- 

^4 

4.  What  is  the  sum  of  14|  and  16|-?  Am.  31f. 

183.  Rule  for  Addition  of  Fractions.  —  i.  Reduce  the  frac- 
tions, if  necessary ,  to  sirnilar  fractions,  add  their  nu- 
merators,  and  write  under  the  sum  the  common  de- 
nonviruitor. 


I'MA  CTIONS.  {S3 

If  there  be  mixed  numbers  or  integers,  add  the  frac- 
tions and  integers  separately ,  and  unite  the  results. 

PltOBLEMS. 

What  is  the  sum  of — 


1.^  and  if?      Ans.lii^. 

2.  if  and  f? 

3.  if  and  T%?      Am.2i^. 
4.^,f_aud3^? 

5.  ^„  ii  and  I?  Ans.  2^-,. 


6.  I,  I,  I  and  f  ?  Ans.  Z\. 

7.  15|,  2|i  and  21  \  ?  Aiis.  45  l|. 

8.  121  24|  and  17f  ? 

9.  6f|,fand7i?     ^ns.  14f|i 
10.  120|i,  11  and22^? 

11.  What  is  the  sum  of  6  +  5|  +  3yV  +  15f  ? 

12.  A  farmer  has  in  one  field  41^^  acres ;  in  a  second,  30| 
acres  ;  and  in  a  third,  60f  acres.  How  many  acres  are  there 
in  the  three  fields  ?  Ans-  1323^.. 


SECTION    XVIII. 

SUBTRACTIOJ^  OF  FliACTIOJTS. 

184. — Ex.  1.  If  you  should  have  7  eighths  of  a  doHar,  and 
should  spend  6  eighths  of  a  dollar,  what  part  of  a  dollar  would 
you  have  left  ? 

2.  John  had  -^  of  a  melon,  and  gave  ^  of  the  melon  to  a 
classmate.     What  part  of  the  melon  had  he  left  ? 

3.  How  many  eighths  are  f  less  f  ?     f  less  ^  ? 

4.  How  much  is  If  less  f  ?     How  much  is  f^ -f  ^  less  |-f? 

185.  Principle. —  Only  fractions  having  a  common  denom- 
inator can  be  subtracted. 

For,  fractions  having  a  common  denominator  are  similar,  and  only 
similar  numbers  can  be  subtracted.  (Art.  57 — 1.) 

WRITTEN  EXERCISES. 

186.— Ex.  1.  What  is  the  difierence  between  -f  and  -^1 

±_J^^  12^ i 12  —  4  _  ^8_  Solution.  —  Reducing      the 

9  27  ~  27  27  ~  27  ~  27  '  given  fractions  to  similar  frac- 
tions, and  finding  the  difference  between  their  numerators,  we  have  2t» 
which  is  the  difference  required. 


84  FRA  CTIOXS. 

2.  What  is  the  diflereuce  between  |4  and  y\  ?     Ans.  ^ff. 

3.  What  is  the  ditference  between  8^  and  5f  ? 

3       rylo  Solution.  —  The  given  fractions  reduced  to 

'^      ^12  12         similar  fractions  give  8|^  =^  8j\,  and  5|  =  5i*j. 

-2  p-  8  Since  -/j  cauuot  be  taken  from  j\,  we  take  1 

O'Z^^  O'TZ  „_  12    e Q 1 : n 1   _jj 


S  12        one,  or  ||,  from  8  ones,  leaving  7  ones,  and  add- 

T\-rc  t7i7         ing  the  \\  to  the  -^^\\aM&\\\  1\%  —  5i*t  =  2t-V, 

•^  v;?        the  dinerence  required. 

4.  What  is  the  difference  between  9|  and  8^  ?       ^?i5.  ||^. 


187.  Rule  for  Subtraction  of  Fractions.— i^etZitce  ;^/2,e  fy^ac- 
tions,  if  necessary ,  to  similar  fractions,  and  write 
the  difference  of  the  numerators  over  the  common  de- 
nominator. 

If  there  he  mixed  numbers,  subtract  first  the  frac- 
tional part  of  the  subtrahend,  and  then  the  integral 
part,  and  unite  the  resitlts. 

PR  OB  Z,  EMS. 

What  is  the  difference  between — 


5.  faud^^?  Ans.  ^. 

6.  J-fiandf? 

7.  14fand96^?    Ans.81^. 

8.  71-1- and  i|? 


1.  i^and^?  Ans.  -^. 

Z.    2-9  auu  2  9  • 

3.  II  and  f?  Ans.  ^. 

4.  T^andf? 

9.  A  boy  had  ff  of  a  bushel  of  chestnuts,  and  sold  f  of  a 
bushel.     What  part  of  a  bushel  had  he  left  ?  Ans.  -fy. 

10.  A  merchant  owned  -^  of  a  ship,  and  sold  f  of  the  ship. 
What  part  of  the  ship  had  he  left  ? 

11.  If  4f  be  taken  from  13,  what  will  be  left? 

13  ^1^1 

o 

^  Solution. — Taking  one  of  the  13  ones  and  calling 

4g        itf,  wehave  13  =  12|. 

J  From  12|  taking  4f,  we  have  8 J,  the  result  required. 

12.  How  much  less  than  800  is  61^7  ^jw.  738^^. 

13.  If  you  should  buy  a  horse  for  200^^  dollars,  and  should 
pay  down  149|  dollars,  how  much  would  you  owe  for  him  ? 


FRACTIONS.  85 


SECTION    XIX. 
MULTIPLICATIOJ^  OF  FRACTION'S. 

CASE   I. 

Fractions  Multiplied  by  Integers. 

188. — Ex.  1.  At  3  tenths  of  a  dollar  each,  what  will  3  slates 
cost  ? 

2.  If  a  man  can  cut  -^  of  a  cord  of  wood  in  1  hour,  how 
much  can  he  cut  in  5  hours  ? 

3.  At  f  of  a  cent  each,  what  will  6  apples  cost  ? 

4.  How  much  is  3  times  y\  of  a  dollar  ?  5  times  ^  of  a 
cord  ? 

5.  How  many  thirds  are  6  times  f  ?  How  many  ones  are 
6  times  f  ? 

6.  How  much  Avill  8  men  earn  at  3^  dollars  a  day  ?  How 
much  at  2f  dollars  a  day  ? 

189.  Principle. — MuUiplying  the  numerator  or  dividing  the 
denominator  of  a  fraction  by  any  number  multiplies  the  fraction 
by  that  number. 

For,  in  the  one  case  the  number  of  parts  is  increased,  while  their  size 
remains  unchanged ;  and  in  the  other  case  the  size  of  the  parts  is  in- 
creased, while  their  number  remains  unchanged. 

mi  6       ri     -i.    .  6X3       18       ^  ,6  ^       z? 

Thus,  -  =  ^;but— =  -  =  6';and— ^=y  =  ^. 

WRITTEN  EXERCISES. 


Solution. — 8  times  H  are 
ff,  which,  reduced,  equals  b\. 

Or,  since  dividing  tlie  de- 
nominator of  a  fraction  also 
multiplies  the  fraction,  8  times 


190.- 

Ex.1. 

Multiply  \l 

by  8. 

fxs 

_  11X8  _88  _  p-   8 
16           16            16 

-H 

Or, 

§xs 

11 

~  16-^& 

~  2    ~     2 

^  are  y,  or  5|,  the  same  result  as  before. 

2.  Multiply  -fl  by  5.  A)i8.  4^. 


86 


FRACTIONS. 


3.  What  is  the  product  of  llf  multiplied  by  8? 
11   X8=88 


-X8=   6 


lljX8=94 

4 


Solution. — Since  llf  equals  11  +  f,  the  product 
of  11|  multiplied  by  8  is  the  same  as  8  times  11  plus 
8  times  f . 

8  times  11  are  88,  and  8  times  |  are  ^5^,  or  6. 

88  and  6  are  94,  the  product  required. 


191.  Rule  for  Multiplying  a  Fraction  by  an  \x\\%%^r.— Multiply 
the  numerat'07\  or  divide  the  denoi^vinator ,  hy  the 
integer. 

If  the  multiplicand  he  a  mixed  numher,  inultiply 
the  integer  and  fraction  separately ,  and  add  the 
products. 

Multiply — 


1.  ii  by  12 

2.  If  by  71 

3.  I  by  63. 


Am.  h\. 


Ana.  35. 


4.  i|  by  13. 

5.  1^  by  100. 

6.  li  by  69. 


10-^ 


A}is.  63. 


7.  What  is  the  product  of  j\  X  19  ?  Ans.  10||-. 

8.  What  will  9  yards  of  cloth  cost  at  5f  dollars  a  yard  ? 

9.  Jason  gathered  5|  bushels  of  apples,  and  Willie  gathered 
6  times  as  many.     How  many  bushels  did  Willie  gather  ? 

10.  If  it  takes  y^  of  a  yard  of  cloth  to  make  one  vest,  how 
many  yards  will  it  take  to  make  24  vests  ? 


Ans.  101 


CASE   II. 


Integ-ers  Multiplied  by  Fractions. 

192. — Ex.  1.  Henry  has  12  apples;  how  many  is  1  third  of 
the  number?     How  many  are  2  thirds  of  the  number? 

2.  John  had  20  cents,  and  gave  away  1  fourth  of  them. 
How  many  cents  did  he  give  away  ?     What  is  f  of  20  cents  ? 

3.  How  much  is  I  of  22  dollars?     |  of  25  dollars? 

4.  How  much  is  |-  of  28  ?    f  of  31  ?    f  of  46  ? 

193.  Principle. — A  number  is  niuUipUed  hy  a  fraction  by 
ohtainbni  such  a  part  of  the  mimber  as  the  fraction  indicates. 

l'"or,  uudtiplyini,'  any  inte,t,'er  by  1  is  taking  it  once;  multiplying  it 
by  5,  is  taking  1  Jiftk  of  it ;  by  I,  is  taking  2  Jiftli.s  of  it,  etc. 


FRACTIONS.  87 

WniTTJEN   EXEItCISES. 

194.— Ex.  1.  Multiply  35  by  f ,  or  fiud  f  of  35. 

5  Solutions.  —  J-  =  f  of   5  ;    lience,   f 

35  X  ~  =  ^^  =  25  times  35  =  ^  of  5  times  35 ;  or  ^^  = 

Or,            '  H^  =  25. 

5  Or,  since  f  :=5  times  ^,  f  of  35^^5 

o  -  ^  -5 ^"  V  /r ^  r  times  i  of  35 :  \  of  35  is  5,  and  5  times 

'        '  5  are  2o. 

2.  Multiply  45  by  |,  or  find  f  of  45.  Ans.  33|. 

195.  Rules  for  Multiplying  an  Integer  by  a  Fraction.— i.  Mul- 
tiply the  integer  by  the  numerator  of  the  multiplier, 
and  divide  the  product  by  the  denominator.     Or, 

2.  Divide  the  integer  by  the  denominator  of  the 
multiplier,  and  multiply  the  quotient  by  the  nume- 
rator. 

PROBLEMS. 

Multiply — 

1.  7  by  f.  Ans.  2^ 

2.  19  by  ^. 

3.  13  by  ^.  Ans.  9^. 

4.  163  by  f 

9.  What  is  the  value  of  136  X  ttV-  ^ns.  44ff. 

10.  What  will  3|-  tons  of  hay  cost  at  $25  per  ton  ? 

(O^       Q        lyp:  Solutions. — If  1  ton  cost  25  dollars,  3  tons 

25X8    =75  ^^11  j,Qgj.  3  (.j^gg  25  dollars,  or  75  dollars,  and  f 

25  X     -=  10  of  a  ton  will  cost  f  of  25  dollars,  or  10  dollars. 

—      75  dollars  and   10  dollars  are  85  dollars,  the 

25  X  3-  =  85  result  required. 

Qj.  Or,  if  1  ton  cost  25  dollars,  3f ,  or  y ,  tons  will 

_  _        17  „  _  cost  y   times  25  dollars,  or  ^  of  25   dollars, 

'^     -^   5    ~  which  is  85  dollars,  the  same  result. 

11.  What  wrll  31|-  tons  of  coal  cost  at  12  dollars  a  ton? 

Am.  382  dollars. 

12.  What  will  f  of  an  acre  of  land  cost  at  120  dollars  per 
acre  ? 


5.  105  by  ^.         Ans.  12. 

6.  121  by  f 

7.  220  by  ^.         Ans.  33. 

8.  100  by  ^. 


88  FMA  CTIONS. 

13.  How  much  must  be  paid  for  17|  hundred-weight  of 
sugar,  at  17  dollars  per  hundred- weight  ? 

14.  If  a  train  of  cars  moves  at  the  rate  of  25  miles  per 
hour,  how  far  will  it  move  in  24|-^  hours  V 

A}i^.  619y^2  miles. 

CASE    III. 

Fractions  Multiplied  by  Fractions. 

196. — Ex.  1.  How  much  is  1  third  of  6  eighths  of  a  dollar? 
HoAV  much  is  2  times  1  third  of  6  eighths  of  a  dollar  ? 

2.  If  you  have  f  of  a  melon,  and  your  brother  has  f  as 
much,  what  part  of  a  melon  has  your  brother  ? 

3.  How  much  is  f  multiplied  by  |  ?     |-  multiplied  by  f  ? 

4.  What  is  i  of  f  ?    What  is  |  of  f  ?     How  much  is  f  mul- 
tiplied by  f  ? 

DEFINITION. 

197.  A  Compound  Fraction  is  a  fraction  of  a  fraction. 

Thus,  I  of  f  is  a  compound  fraction.     The  word  of  in  the  expression 
denotes  multiplication. 

WRITTEN  EXEItCISES. 

198.— Ex.  1.  Multiply  f  by  f ,  or  find  f  of  |. 

8  s       2A         8  SoiiTJTiONS. — f  of  f  is  the  same  as  3  times 
<)'^7'~6^'~'n              I  of  f ;  f  of  f  is  ^^,  or  ^\,  and  3  times  ^\ 

Or,  are  ^^^,  or  f  f ,  which  reduced  is  ^\,  the  result 

8^S_  sy.-S 8        required. 

9  7       ^X?        21  Or,  indicating  the  multiplication  and  can- 

^  celling,  we  have  ^^i  ^^  before. 

2.  Multiply  II  by  ^\,  or  find  ^  of  ||. 

3.  What  is  the  product  of  8f  by  4^? 

^         I       20        '■'I  Solution. — 8f  is  equal  to   -{\  and  4^  is 

^ J  X  .^j"  =^  -^  X  ^  equal  to  V  ;  hence,  8  j  X  4i  is  the  same  as  '^ 

,oa             o  X  V.     Cancelling  and  multiplving,  we  have 

'^  'J'  —  *^^i  -^s^j  which  reduced  is  36f ,  the  result,  required. 


FRA  CTIONS. 


89 


199.  Rule  for  Multiplication  of  VvdicWon^.  —  Multiply  the 
nuiyveratoTS  together  for  the  numerator,  and  the  de- 
nominators for  the  denominator,  of  the  product. 

If  there  be  mixed  numbers,  reduce  them  to  frac- 
tions before  multiplying. 

The  value  of  a  Compound  Fraction  is  found  by  performing  the  multi- 
plication indicated  by  the  expression. 


Multiply- 

12    "j    3  5- 


I  by  -f . 
Hbyf 


by 


FJtOBLJSMS. 


33 

To- 


Ans.  -i-i- 


5. 
6. 

7. 


-2-  bv  ^ 


by  tVtt- 


^by 

2 


4  5    "J     2  1- 

by^J-. 


Ans. 


T5- 


9. 
10. 
11. 
12. 
13. 


of 


What  is  the  value  of 

What  is  the  value  of  |  of  f  ? 

Multiply  17f  by  14|. 

Two  factors  are  79^  and  9\. 


1  3  9 

1  7  • 


65 


^)is.  255f. 
What  is  their  product  ? 


A  boy  gathered  ^  of  a  bushel  of  berries,  and  his  brother 
gathered  ^  as  many.     What  quantity  did  his  brother  gather  ? 

14.  How  many  yards  are  there  in  12f  pieces  of  cloth,  each 
containing  26^  yards  ? 

15.  What  will  192^  tons  of  coal  cost  at  8f  dollars  per  ton  ? 

16.  At  y^  of  a  dollar  a  yard,  what  will  y^g-  of  a  yard  of 
cloth  cost? 

17.  If  a  family  use  If  barrels  of  flour  in   1   month,  how 


many  barrels  will  it  use  in  114  months? 


Ans. 


18|. 


18.  If  you  should  buy  a  watch  for  80f  dollars,  and  sell  it 
for  f  of  the  cost,  for  how  much  would  you  sell  it  ? 

19.  At  the  rate  of  2f  tons  per  acre,  how  much  hay  can  be 
obtained  from  21-^  acres  ?  Ans.  50f  tons. 

20.  If  a  man  can  build  |^  of  a  wall  in  a  day,  what  part  of 
it  can  he  build  in  -f-  of  a  day  ? 

21.  What  is  the  value  of  f  of  a  bushel  of  clover-seed  at 
7y^  of  a  dollar  per  bushel  ?  Ans.  6^  dollars. 

of 


22.  What  is  the  value  of  f  of 


iof 


If  of  4?    Ans.^. 


90  FRACTIONS. 

SECTION    XX. 
BIVISIOK  OF  FBACTIOJ^S. 

C^SE    I. 

Fractions  Divided  by  Integers. 

200. — Ex.  1.  Charles  divided  4  fifths  of  a  melon  between 
his  two  brothers.    What  part  of  the  melon  did  he  give  to  each  ? 

2.  If  ^  of  a  ship  is  owTied  by  5  men  in  equal  shares,  what 
part  of  the  ship  is  each  man's  share  ? 

3.  If  f  of  an  acre  of  land  be  divided  into  two  equal  house- 
lots,  what  part  of  an  acre  will  each  lot  be  ? 

201.  Principle. — Dividing  the  numerator,  or  multiplying  the 
denominator,  by  any  number,  divides  the  fractimi  by  that  number. 

For,  in  the  one  case  the  number  of  parts  is  diminished,  while  their 
size  remains  unchanged ;  and  in  the  other  case  the  size  of  the  parts  is 
diminished, while  their  number  remains  unchanged. 

Thus,  -  =  2;  but  —  =  j;  and  — ^  =  ^=-. 
rVRITTEX  EXERCISES. 

202.— Ex.  1.  Divide  |f  by  5. 

Solutions. — Since  dividing  the  nume- 
rator of  a  fraction  divides  the  fraction,  -^ 
divided  by  5  gives  j^^,  the  result  required. 

Or,  since  multiplying  the  denominator 
of  a  fraction  divides  the  fraction,  la  -=-  5  = 

I  ^g,  or,  by  reduction,  fj,  as  before. 

2.  Divide  31^  by  5. 

o-jA^  ^  =  — ^  ^  =  —  =  /?-  Solutions. — 31^  is  equal  to 

^^^  -^  4    ~    4        l|^;   hence,    31Jh-5    is   the 

same  as  if  ^  -^  5 ;  and  i|^  -r-  5 
is  \^,  or  6^. 

Or,  we  may  divide  without 
6-7  reduction.      Thus,   5   is    con- 

tained in  31  \,  6  times  with  a 
remainder  of  Ij,  wliich  is  ecjual  to  :] ;  5  is  contained  in  \,  \  time;  and 
G  plus  \  is  Q\,  the  same  result  as  before. 


10 

11 

^5  = 

10^5 
11     ~ 

2 
11 

Or, 

2 

10 

11 

^5  = 

IIX^^ 

2 
11 

^^*  5)3lj 

4 


FRACTIONS. 


91 


3.  Divide  II  by  12. 


W  by  19. 


203.  Rule  for  Dividing  a  Fraction  by  an  \xiieq^r.— Divide  the 
numerator,  or  multiply  the  denoTYiinator ,  by  the  in- 
teger. 

If  the  dividend  is  a  mixed  number,  reduce  it  to  an 
improper  fraction  before  dividing ;  or,  divide  the  in- 
teger and  fraction  separately,  and  unite  the  results. 


Divide — 

PROBLEMS. 

1. 

2. 
3. 
4. 

f  by6. 
|by4. 
i|by5. 
W  by  12. 

Ans.  \. 
Ans.  If. 

5.  ^  by  12. 

6.  y  by  51. 

7.  ff  by  45. 

8.  V  by  27. 

Am.  ^ 
Ans.  yV 

9.  If  11  yards  of  cloth  cost  16|  dollars,  liow  much  does 
1  yard  cost  ?  Ans.  1|  dollars. 

10.  What  is  the  quotient  of  |  divided  by  9  ? 

11.  What  is  the  value  of  ^^  ^  20  ? 
What  is  the  value  of  22f  --  7  ? 
Divide  ff^  by  100. 


27 
32- 


Ans. 


■5"ir9- 


12. 
13. 

14.  A  farmer  raised  91|-|  bushels  of  wheat  upon  5  acres 
of  land.     How  many  bushels  was  that  per  acre  ? 


C^SE    II. 

Integers  Divided  by  Fractions. 

204. — Ex.  1.  How  many  apples,  at  f  of  a  cent  each,  can  be 
bought  for  6  cents  ? 

2.  How  many  times  2  thirds  of  a  cent  in  18  thirds  of  a 
cent  ?     How  much  is  ^  of  3  times  6  ? 

3.  When  coffee  is  f  of  a  dollar  a  pound,  how  many  pounds 
can  be  bought  for  6  dollars  ? 

4.  How  many  times  is  f  contained  in  6?     In  12? 

5.  If  meal  is  |^  of  a  dollar  per  bushel,  how  many  bushels 
can  be  bought  for  14  dollars  ? 

6.  How  many  times  is  |  contained  in  14  ?     In  21  ? 


92  FRACTIONS. 

WniTTEN  EXETtClSES. 

205.— Ex.  1.  Divide  15  by  f. 

S       60       3  Solutions. — 15  is  equal  to  ^f  ;  and 

1^  ~^  A  ^  ~r  ~^4  ^^  ^^  fourths  divided  by  3  fourtlis  gives 

Or,  20,  the  result  required. 

5  Or,  since  15^-1  is  15,  15-=-^  must 

15  ^^  =  ^^^^  =  20  be  4  times  15,  and  15H-f  must  be  ^ 

*  of  4  times  15,  or  20,  the  same  result. 

2.  Divide  63  by  f.  Ans.  105. 

206.  Rule  for  Dividing  an  Integer  by  a  Fraction.— ^ec^z^ce  the 
dividend  to  a  fraction  similar  to  the  divisor,  and  di- 
vide the  numerator  of  the  dividend  by  the  numerator 
of  the  divisor.     Or, 

Multiply  the  integer  by  the  denominator  of  the 
divisor,  and  divide  the  result  by  the  numerator. 

PJtOBLEMS. 


Divide — 

1.  16  by  f. 

2.  19  by  |.  Am.  2^. 

3.  54  by  f 

4.  100  by  |i.         Ans.  129^. 

9.  Divide  40  by  5f. 


5.  65  by  f 

6.  29  by  |.  Am.  38f . 

7.  llOby^^j. 

8.  144  by  L|.  Am.  156. 


Solution. — 5f  equals  ^^  ; 
(.^__27^,     ,^.    37       40  X  5  11     40 -^  5§  is  the  same  as  40  H- 

^5~5'    -^  -^  ~  5    ~      27      ~  '27       V)  which  gives  1\\,  the  re- 
sult required. 

10.  Divide  97  by  ^.  Am.  \1\^. 

11.  Divide  365  by  365^. 

12.  At  8f  dollars  a  ton,  how  many  tons  of  coal  can  be 
bought  for  63  dollars?  Ans.  ''i\. 

13.  At  ^  of  a  dollar  a  yard,  how  many  yards  of  cloth  can 
be  bought  for  27  dollars  ? 

14.  Smith's  farm  consists  of  133  acres,  and  is  divided  into 
fields  of  16f  acres  each.     How  many  fields  are  there? 


FRACTIONS.  93 

CASE    III. 

Fractious  Divided  by  Fractions. 

207. — Ex.  1.  Into  how  many  parts,  of  3  tenths  each,  can  you 
divide  9  tenths  of  an  orange  ? 

2.  How  many  times  is  y\  of  a  melon  contained  in  -j^  of  a 
melon  ? 

3.  How  many  sixths  are  -|  ?     How  many  times  ^  in  |  ?    \ 
in|? 

4.  How  many  pounds  of  coffee,  at  f  of  a  dollar  a  pound, 
can  be  bought  for  f  of  a  dollar  ? 

5.  How  many  times  f  in  f  ?    -f  in  f  ?    f  in  ^  ? 

DEFINITIONS. 

208.  A  Complex  Fraction  is  a  fraction  having  a  fraction  in 
one  or  both  of  its  terras. 

3 

Thus,  -2-  is  a  complex  fraction,  and  is  an  expression  of  division  of 
fractions. 

209.  A  fraction  is  inverted  when  the  denominator  is  taken 
for  the  numerator,  and  the  numerator  for  the  denominator. 

Thus,  I  inverted  is  | ;  f  inverted  is  I. 

WMITTEN   KXEJiCISES. 

210.— Ex.  1.  Divide  |  by  |. 

4:_^^_S6__^10^_  e)6_  ___  ^5  SOLUTIONS.  —  Expressed     as 

5       9       45        45           10           5  similar  fractions,  f  -^  |  is  the 

Or,                       5  ■  s^™«^  ^  If  -^  H ;  Sfi  forty-fifths 

4  2       ^-XQ       18        cjS  divided  by  10  forty-fifths  gives 

5  ~  9       5  y.-2        5  5  3y^^,  or  3f ,  the  result  required. 

Or,  since  |  -^- 1  is  |,  f  h-  ^ 
must  be  9  times  |,  or  ^^,  and  4  h-  |  must  be  |^  of  9  times  |,  or  f^, 
which,  reduced,  is  3f,  the  same  result. 

In  the  last  solution  the  division  is  performed  by  multiplying  by  the 
divisor  inverted. 

2.  What  is  the  quotient  of  -^^  divided  by  ^  ?       Ans.  f , 

3.  What  is  the  quotient  of  \^  divided  by  ^  ? 


94 


FRACTIONS. 


211.  Rules  for  Dividing  a  Fraction  by  a  Fraction.— i.  Reduce 
the  divisor  and  dividend,  if  necessary,  to  similar 
fractions,  and  divide  the  ninnerator  of  the  dividend 
Jjy  the  numerator  of  the  divisor.     Or, 

2.  Multiply  the  dividend  by  the  denominator  of  the 
divisor,  and  divide  the  result  by  the  numerator.     Or, 

3.  Multiply  the  dividend  by  the  divisor  inverted. 
The  value  of  a  complex  fraction  is  found  by  performing  the  division 

indicated  by  the  expression. 


Divide — 

1.  f  by  f. 

2.  ^hyli. 

3.  f  by  |. 

4.  H  by  ^.^ 

^-  Too"  by  looo- 
6.  iibyf. 


PROBLEMS. 


Ans.  ff. 


Am.  1^. 


Ans.  1||. 


9. 
10. 
11. 
12. 


Abyf 
fby^. 
Hby:^. 

To  "J   9- 

2  1    l^v    14 

T3"  '^y  T5- 

5  2  Ky    1 

looooo "y  1 


13.  What  is  the  value  of  |f  - 1? 

3. 

14.  What  is  the  value  of  ^? 


Am.  2h 


Ans.  f^ 


J-.    -4  ns.  2-5^, 
Ans.  1^2- 


3 

•?   X  T? 

■4  K  5  ' 
2 


0 
10 


SoLtJTiON.  —  |-  is  the  same  as  f  h-  |, 

6" 

and  I  -f-  f  gives  f'g-,  tlie  result  required. 


15.  What  is  the  value  of  ^  ? 

t 

16.  What  is  the  value  of  ^  ? 

17.  Divide  40^  by  |. 


Ans.  II 
Ans.  ||- 


SI 


SI 


04 


Solution .—40^  =  y  ; 
hence,  40^  -^  |  is  tlie  same  as 
V  -=-  I,  which  gives  54,  the 
result  required. 

18.  How  many  times  is  ^  contained  in  5f  ?         Ans.  5^. 

19.  What  is  the  value  of  174  -  1^4? 

20.  Divide  \^  by  7i.  Ans.  ^. 


FRA  CTIONS.  95 


3 


21.  -^  is  equal  to  what  number?  Ans.  -^. 

22.  If  you  should  spend  |  of  a  dollar  a  w^ek,  in  Avhat  time 
would  you  spend  7^  dollars?  Ans.  11|  weeks. 

23.  If  I  pay  ^  of  a  dollar  for  f  of  a  bushel  of  corn,  what 
shall  I  pay  for  a  bushel  ? 

24.  If  a  man  can  travel  llf  miles  in  If  hours,  in  what  time 
can  he  travel  one  mile  ? 

25.  When  5^  pounds  of  veal  cost  73f  cents,  what  is  the 
cost  per  pound  ?  Ans.  14^  cents. 


SECTION  XXI. 

RELATION  OF  JS' UMBERS. 

212.— Ex.  1.  What  part  of  7  cents  is  1  cent?  Is  2  cents? 
Is  3  cents  ? 

2.  How  much  is  1  divided  by  7  ?  2  divided  by  7  ?  3  di- 
vided by  7  ? 

3.  What  part  of  2  oranges  is  f  of  an  orange  ? 

4.  AVhat  part  of  2  is  1  f    What  part  of  2  is  |  of  1  ? 

DEFINITION. 

213.  The  Relation  of  one  number  to  another  is  the  part  the 
former  is  of  the  latter,  or  is  the  number  of  times  the  former 
contains  the  latter. 

214.  Principle. —  Only  similar  numbers  can  have  relation  to 
each  other. 

For,  only  such  numbers  as  have  the  same  unit  can  be  compared  with 
or  measured  by  each  other. 

WHITTEN  EXEMCTSES. 

215.— Ex.  1.   What  is  the  relation  of  21  to  7? 

^  SoLTTTiON. — Tlie  relation  of  21  to  7  is  the  num- 

21to7  =  ~  =  3      ber  of  times  that  21  contains  7. 

21  is  3  times  7 ;  hence  3  is  the  relation  required. 


96  FRACTIONS. 


2.  What  is  the  relation  of  f  to  4,  or  what  part  of  4  is  |  ? 

o  Q  I  Solutions.— The  relation  of  |  to 

to  4  =  1^-^  4=  JT'^—  ^        4  is  f  divided  by  4,  or  i. 

Or,  since  1  is  ^  of  4,  \  is  ^  of  ^ 


8X4- 
Or, 


2    _  1 
3X  A~  6  of  4,  which  equals  \  of  4. 

3.  What  is  the  relation  of  |  to  f ,  or  what  part  of  f  is  |  ? 

Solutions.— The  relation  of  f 

^fa-  =  -^^  =  ^^  =  ^       to  I  is  I  divided  bv  f ,  which   is 
5     "^  4        5    '   4        5X3       15         2ji^  ^^    8 

Or  ^^'  ^'"^®  ^  i*^  I.  I  is  f  of  f,  i  of 

?2<i  ^  J_  1  is  I  of  f  of  f,  and  |  of  1  are  |  of 


5X3       15 


I  of  f ,  or  ^3  of  f ,  as  before. 


4.  What  is  the  relation  of  1|  to  y^y?  J.n5.  Sy^^j. 

216.  Rules  for  Finding  tlie  Relation  of  Numbers.— i.  Divide 
the  number  compared  by  that  with  which  it  is  com- 
pared.    Or, 

2.  Divide  the  number  denoting  the  part  by  that  de- 
noting the  whole. 

FItOBT.  EMS. 

What  is  the  relation  of —  What  part  of — 

1.  31  to  17?  Ans.  Iff. 

2.  42tof? 

3.  If  to  14?  Ans.^. 

4.  I  to  II? 


5.  35  is  21  ?  Ans.  |. 

6.  36  is  ^  ? 

'  •  2  5   ^°  TT  •  Ji.nS.  YTz- 

8.  7|  is  f  ? 


9.  James  had  100  dollars,  and  gave  away  7|  dollars.    What 
part  of  100  dollars  did  he  give  away?  Ans.  -^. 

10.  A  man  gave  125  dollars  for  a  carriage,  and  sold  it  for 
100  dollars.     For  what  part  of  the  cost  did  he  sell  it  ? 

11.  If  James  receives  60  dollars  for  15  Aveeks'  Avork,  how 
much  should  he  receive,  at  the  same  rate,  for  25  weeks'  Avork  ? 

20  Solution.— If  he  receives  60  dollars  for 

«5_  _  5  ^   -e&X  5  _  jQQ     If)  weeks,  for  2r-,  weeks,  which  is  f  ^,  or  f,  as 
1-5        S  -»■  many  weeks,  he  should  receive  f  of  60  dol- 

lars, or  100  di)ll;irs. 


FE  ACTIONS.  97 


12.  If  a  man  can  hoe  a  field  in  3f  days,  Avhat  part  of  it  can 
he  hoe  in  2^  days  ?  Ans.  f . 

13.  A  and  B  hire  a  pasture  together ;  A  puts  in  15  cows, 
and  B  puts  in  40.  If  B's  share  of  the  rent  is  200  dollars,  how 
much  is  A's  ? 

14.  Jason  gathered  f  of  a  bushel  of  nuts,  and  Daniel 
gathered  f  of  a  bushel.  What  is  the  value  of  Daniel's,  ii 
Jason's  are  worth  150  cents  ? 

15.  Jones  has  175  sheep,  and  Eeed  has  f  as  many.  How 
would  their  numbers  compare  if  Reed  were  to  have  135  more? 

Ans.  Reed  would  then  have  f  as  many  as  Jones, 


TEST  QUESTIONS. 

217. — 1.  What  kind  of  fractions  only  can  be  ADDED ?  Why?  How 
are  dissimilar  fractions  prepared. for  adding?  How  may  mixed  numbers 
be  added  ?     Give  the  rules  for  addition  of  fractions. 

2.  Why  is  it  necessary  that  fractions  should  have  a  common  denomi- 
nator before  subtracting  ?  How  may  mixed  numbers  be  subtracted  ? 
Give  the  rules  for  subtraction  of  fractions. 

3.  How  is  a  fraction  multiplied  by  an  integer  ?  Why  does  multi- 
plying the  numerator  multiply  the  fraction  ?  Why  does  dividing  the 
denominator  multiply  the  fraction  ?  How  is  a  number  multiplied  by  a 
fraction  ?     Why  ? 

4.  What  is  the  rule  for  multiplying  a  fraction  by  an  integer?  The 
rules  for  multiplying  an  integer  by  a  fraction  ?  When  the  multiplier  is 
an  improper  fraction,  will  the  product  be  greater  or  less  than  the  multi- 
plicand ?    Why  ? 

5.  What  is  a  compound  fraction  ?  How  do  you  find  the  value  of 
a  compound  fraction  ?  What  are  the  rules  for  multiplying  a  fraction  by 
a  fraction  ? 

6.  What  is  a  complex  fraction  ?  How  is  a  fraction  inverted  ?  How 
is  the  value  of  a  complex  fraction  found  ? 

7.  In  what  two  ways  may  a  fraction  be  divided  ?  Why  ?  V*''hat  is 
the  rule  for  dividing  a  fraction  by  an  integer  ?  What  are  the  rules  for 
dividing  an  integer  by  a  fraction  ?  The  rules  for  dividing  a  fraction  by 
a  fraction  ? 

8.  What  is  the  relation  of  one  number  to  another?  What  numbers 
only  can  have  relation  to  each  other  ?  Why  ?  Give  the  rules  for  finding 
the  relation  of  numbers. 


98  REVIDW  PROBLEMS. 

SECTION    XXII. 
REVIEW   PROBLEMS. 

MJENTA.L   EXERCISES. 

218. — Ex.  1.  How  many  apples  are  there  iu  ^,  \,  |  and  yV 
of  an  apple  ? 

2.  How  much  is  -^  less  than  f  ?     \^  more  than  |-  ? 

3.  Add  I,  I  and  ^-f    f ,  f  and  ^. 

4.  Arthur  has  5f  dollars,  and  John  has  9|  dollars.     What 
is  the  total  amount  of  their  money  ? 

5.  I  bought  a  hat  for  3^^  dollars,  and  gave  in  payment  a 
five-dollar  bill.     How  much  change  should  I  receive  ? 

6.  If  you  should  earn  in  a  week  7|-  dollars,  and  sjiend  5f 
dollars,  how  much  would  you  save  ? 

7.  At  5|-  dollars  each,  how  much  will  11  hats  cost  ? 

8.  A  man  owned  -ff  of  a  ship,  and  sold  |  of  his  share. 
What  part  of  the  ship  did  he  sell  ? 

9.  A  farmer  has  54  sheep,  and  his  son  has  f  of  that  number. 
How  many  has  his  son  ? 

10.  I  sold  a  watch  for  48  dollars,  which  was  f  of  its  value. 
What  was  its  value  ? 

11.  The  difference  between  |  and  f  of  my  age  is  4  years. 
What  is  my  age  ? 

12.  After  spending  i  of  f  of  my  money  I  had  22  dollars  left. 
How  much  did  I  have  at  first  ? 

13.  If  a  boy  can  pick  |f  of  a  bushel  of  berries  in  5  hours, 
what  part  of  a  bushel  can  he  pick  in  1  hour  ? 

14.  How  many  yards  of  cloth,  at  f  of  a  dollar  a  yard,  can 
be  bought  for  6  dollars? 

15.  At  ^  of  a  dollar  a  peck,  how  many  pecks  of  apples  can 
be  bouglit  for  ^  of  a  dollar  ? 

16.  How  many  months  will  5f  barrels  of  flour  last  a  family 
which  uses  f  of  a  barrel  a  month  ? 

17.  A  boy  being  asked  how  much  money  he  had,  said  that 
I  of  I  of  it  was  4^-  dollars.     How  much  did  he  have  ? 


REVIE\r  PROBLEMS.  99 

18.  How  many  times  can  you  pay  10  dollars  from  your 
money,  if  18  dollars  is  f  of  it? 

19.  John  bought  a  horse  for  125  dollars,  and  sold  it  for  f 
of  what  he  gave.     How  much  did  he  gain  by  the  transaction  ? 

20.  If  24  dollars  is  f  of  your  money,  how  many  books  at 
2\  dollars  each  can  you  buy  ? 

21.  Mary  had  28  cents,  and  gave  f  of  it  for  3  oranges.  How 
much  did  each  orange  cost  ? 

22.  If  3  yards  of  cloth  cost  7-^  dollars,  what  wull  ^j  of  a 
yard  cost  ? 

23.  If  13^  tons  of  hay  will  last  22  horses  one  month,  how 
many  tons  will  last  15  horses  the  same  time  ? 

24.  When  butter  is  f  of  a  dollar  a  pound,  how  many  pounds 
of  butter  will  pay  for  7  pounds  of  tea,  worth  y^^-  of  a  dollar  a 
pound  ? 

25.  "When  an  acre  of  ground  is  Avorth  60  dollars,  what  part 
of  an  acre  can  be  bought  for  15  dollars? 

26.  If  I  have  f  of  |  of  18  dollars,  and  you  have  f  of  15 
dollars,  what  part  of  my  money  is  the  difference  of  our 
money  ? 

27.  If  f  of  a  bushel  of  oats  be  given  for  |  of  a  bushel  of 
corn,  what  is  the  cost  of  a  bushel  of  oats  Avhen  corn  is  f  of  a 
dollar  a  bushel  ? 

28.  How  many  pairs  of  shoes,  at  If  dollars  a  pair,  may  be 
exchanged  for  4  pairs  of  boots,  at  44  dollars  a  pair  ? 


WHITTEN  EXERCISES. 

219. — Ex.  1.  Which  is  the  greatest  and  which  the  least  of 
the  fractions  ^,  -^  and  -f^  ?        Ans.  -^-^,  greatest ;  ^,  least. 

2.  A  owns  ^  of  a  steamboat,  B  owns  \,  C  OAvns  \,  and  D 
owns  the  rest.     What  part  of  the  steamboat  does  D  own  ? 

3.  The  sum  of  two  numbers  is  13#,  and  one  of  the  numbers 


is  5|i.     What  is  the  other  number  ?  Ans.  7f|-. 

4.  The  difference  between  two  numbers  is  ^,  and  the  smaller 
of  the  numbers  is  ^.     What  is  the  larger  number? 

5.  What  number  divided  by  |-|  will  give  f  ? 


100  REVIEW  PROBLEMS. 

6.  If  f  of  my  money  is  3612  dollars,  how  much  money 
have  I?  J. ns.  4128  dollars. 

7.  At  Tf  dollars  a  ton,  what  part  of  a  ton  of  coal  can  be 
bought  for  2^  dollars  ? 

8.  What  is  the  greatest  common  divisor  of  1^,  f ,  and  2f  ? 

lA    -     9- —    —     ^  Solution. — 1J,|,2|,  expressed  as  sini- 

2     4        5       20      20      20       ilar  fractions,  having  the  l&ast  common 
Greatest  Com.  Divisor,  /-     denominator,  are  ff,  i|-,  ff.   ^ 

The  greatest  common  divisor  of  the 
numerators  of  these  fractions  is  3 ;  and  the  greatest  common  divisor  of 
the  denominators  is  20.  Hence,  /q  is  the  greatest  common  divisor 
required. 

9.  What  is  the  width  of  the  largest  blocks  that  will  exactly 
fit  across  each  of  two  walks,  the  one  4|-  feet  wide,  and  the 
other  6f  feet?  ^ns.  2-L  feet. 

10.  At  the  rate  of  5^  miles  per  hour,  what  time  will  be  re- 
quired to  walk  126  miles  ? 

11.  A  and  B  own  a  lot;  A  owns  |-  of  it,  and  B  owns  f. 
The  part  that  B  owns  is  worth  30  dollars  more  than  that  owned 
by  A.     How  much  is  each  of  their  parts  worth  ? 

Ans.  A's,  45  dollars ;  B's,  75  dollars. 

12.  What  is  the  least  common  multiple  of  f,  f,  and  -^7 

2  3     J^^J_     _9_     J_  SoLtrxiON.— I,   f,    iV,  expressed   as 

3  4  12  12  12  12  similar  fractions,  having  the  Imzt  com- 
Least  Com.  Mult  =  —  =  6"      "^o"  denominator,  are  -f-^,  ^■^,  J^.     The 

least  common  multiple  of  the  numera- 
tors 8,  9  and  1  is  72 ;  the  least  common  multiple  of  the  denominators  is 
12.    Hence,  ^f,  or  6,  is  the  least  common  multiple  required. 

13.  What  is  the  least  number  of  cents  that  will  pay  for  a 
number  of  peaches  at  |  of  a  cent  each,  a  number  of  apples  at 
f  of  a  cent  each,  and  a  number  of  pears  at  f  of  a  cent 
each  ? 

14.  When  f  of  a  ton  of  coal  costs  6|  dollars,  how  much 
does  I  of  a  ton  cost?  Ans.  6|  dollars. 

15.  A  pole  was  broken  into  two  pieces.  One  of  the  pieces, 
which  measured  31^  feet,  was  f  of  the  length  of  the  pole.  How 
long  was  the  pole  ? 


BE  VIEW    PROBLEMS.  101 

16.  I  have  three  rooms,  13|^  feet,  15|  feet,  and  18  feet  wide, 
respectively.  What  is  the  widest  carpeting  that  will  cover 
each  with  whole  breadths  ? 

17.  John  has  164  dollars,  which  is  ^  as  much  as  Willard 
has.    How  much  has  Willard  ?  Ans.  225|^  dollars. 

18.  If  f  of  an  acre  of  land  cost  48  dollars,  what  will  -^  of 
an  acre  cost  ? 

Solution.— I  of  an  acre  is  -^^  of  an  acre. 
3        6     -tsy-  7  ^^  T5  ^^  ^"  ^cre  cost  48  dollars,  y^^  of  an  acre, 

J  =  ■^Z  '^     -=  o^tJ       or  I  as  much,  will  cost  ^  of  48  dollars,  or  56 
dollars. 

19.  26-|  is  f  of  what  number? 

20.  If  f  of  a  sum  of  money  is  220  dollars,  how  much  is  -^ 
of  it  ?  Ans.  90  dollars. 

21.  When  3|-  cords  of  wood  can  be  bought  for  25  dollars, 
how  much  can  be  bought  for  10|-  dollars? 

22.  If  a  man  inherits  f  of  an  estate  of  2000  acres,  and  sells 
f  of  his  share,  how  many  acres  does  he  retain  ?     Ans.  285f . 

23.  James  bought  a  horse  and  carriage  for  400  dollars.  The 
horse  cost  f  of  the  price  of  the  carriage.  What  was  the  price 
of  each? 

24.  If  5  be  added  to  each  term  of  the  fraction  f ,  how  much 
will  the  value  of  the  fraction  be  diminished  ?  Ans.  -^. 

25.  If  8  men  can  do  a  piece  of  work  in  |-  of  -f-  of  \^  of  o|- 
days,  how  many  men  could  do  it  in  1  day  ?  Ans.  63. 

26.  An  estate  worth  $6000  has  been  divided  as  follows : 
The  original  owner  had  5  children,  each  of  his  children 
had  4  children,  and  his  grandchildren  now  own  the  estate  in 
equal  shares.  What  fraction  represents  the  share  of  each 
grandchild,  and  what  is  the  value  of  15  of  the  shares? 

Ans.  i  of  i  of  $6000 ;  $4500. 

27.  A  man,  whose  estate  was  worth  5000  dollars,  bequeathed 
to  his  widow,  \  of  it ;  to  his  son,  ^  of  it ;  and  the  remainder 
in  equal  shares  to  his  two  daughters.  What  was  the  value  of 
the  share  of  each  ? 

Ans.  Widow,  1666|  dollars ;  the  son,  1500  dollars; 
and  each  daughter,  91 6f  dollars. 
9* 


102  DECIMAL  FRACTIONS. 

SECTION   XXIII. 

DECIMAL  FBACTIOJ^S. 

220. — Ex.  1.  If  a  melon  be  divided  into  ten  equal  parts, 
what  part  of  the  whole  will  one  of  the  pieces  be  ?  Two  of  the 
pieces  ? 

2.  What  is  the  fractional  unit  of  a  number  of  tenths  ? 

3.  If  a  tenth  of  a  melon  be  divided  into  ten  equal  parts, 
what  part  of  the  melon  will  one  of  the  pieces  be  ?  Two  of  the 
pieces  ? 

4.  In  a  number  of  hundredths,  what  is  the  fractional  unit  ? 

5.  If  a  hundredth  of  a  unit  be  divided  into  ten  equal  parts, 
what  part  of  the  unit  will  one  of  the  parts  be? 

6.  In  a  number  of  thousandths,  what  is  the  fractional  unit  ? 

7.  How  many  thousandths  is  a  hundredth?  How  many 
hundredths  is  a  tenth  ?     How  many  tenths  is  1  ? 

8.  How  is  one  tenth  written  as  a  common  fraction  ?  One 
hundredth  ?     One  thousandth  ? 

9.  How  are  two  tenths  written  as  a  common  fraction  ?  Four 
hundredths  ?     Five  thousandths  ? 


DEFINITIONS. 

221.  A  Decimal  Fraction  is  a  number  of  tenths,  hundredths 
or  thousandths,  etc. 

Thus,  1  tenth,  3  tenths,  7  thousandths,  are  decimal  fractions. 

Decimal  fractions  are  so  called  because  they  are  decimal 
divisions  of  a  unit. 


Thus,  if  i  he  di 

vidcd 

into  10  1 

equal  parts, 

each  part : 

is  1  tenth. 

1 

u       — 

10 

" 

"    10 

(( 

" 

1  liundredth. 

1 

«       — 

100 

(1 

"    10 

(( 

ti 

1  thousandth. 

1 

u 

1000 

(1 

"    10 

« 

(1 

1  ten-thousandth, 

And  so  on. 

DECIMAL  FRACTIONS.  103 

222.  Since,  in  the  decimal  system  of  notation,  a  figure  of  any 
order  expresses  -^-^  the  value  expressed  by  the  same  figure 
written  in  the  next  order  on  the  left,  each  figure  written  at  the 
right  of  the  decimal  point  must  express  a  value  y  „  as  great  as 
tliat  expressed  by  the  same  figure  written  in  the  next  order  on 
the  left. 


e,  —  mav  be 
10 

written, 

,0.1, 

or 

simply, 

J. 

1 

n 

100 

a 

0.01, 

li 

.01. 

1 
1000 

a 

0.001, 

a 

.001. 

1 

10000 

ti 

0.0001, 

a 

.0001. 

And  so  on,  in  continuation  of  the  notation  of  integers,  to  the 
right  of  ones,  since  decimal  fractions  and  integers  have  the 
same  scale, 

223.  The  Decimal  Point  (.)  distinguishes  the  decimal  fraction 
from  an  integer. 

224.  A  Decimal  is  a  decimal  fraction  expressed  without  its 
denominator. 

225.  A  Piire  Decimal  consists  of  a  decimal  only. 
Thus,  .85,  which  is  read  35  hundredths,  is  a  pure  deciraa!. 

226.  A  Mixed  Decimal  consists  of  an  integer  and  a  decimal. 
Thus,  15.7,  which  is  read  15  ones  and  7  tenths,  or  15  and  7  tenths,  is  a 

mixed  decimal. 

227.  A  Complex  Decimal  consists  of  a  decimal  with  a  com- 
mon fraction  annexed. 

Thus,  .OOof,  which  is  read  5f  thousandths,  is  a  complex  decimal. 

228.  The  Jfames  of  decimal  orders  are  derived  froin  the 
names  of  the  orders  of  integers. 

Thus,  the  name  of  the  first  order  of  decimals  below  ones  is  tenths,  cor- 
responding; to  tens,  the  name  of  the  first  order  higher  than  ones;  the 
second  order  below  ones  is  hundredths,  corresponding  to  hundreds,  the 
second  order  higher  tlian  ones;  and  so  on.  That  is,  the  names  of  the 
orders  below  and  aljove  ones  correspond  to  each  other,  with  the  exception 
of  the  decimal  termintition  ths. 


104  DECIMAL  FRACTIONS. 

TABLE  OF  OKDEHS. 


76     5     4321        .234     5      67 


INTEGERS.  DECIMALS. 

The  number  in  the  table  is  read :  seven  million  six  hundred 
fifty-four  thousand  three  hundred  twenty-one,  and  two  hun- 
dred thirty-four  thousand  five  hundred  sixty-seven  millionths. 

229.  Principles. — 1.  Ten  of  any  order  o/  decimals  are  equal 
to  one  of  the  order  next  higher. 

For,  decimals  are  a  continuation  of  the  notation  of  integers,  below 
units  or  ones,  by  orders  of  tenths,  hundredths,  etc.,  on  a  scale  of  ten. 

2.  The  denominator  of  a  dccinial  is  1  ivifh  as  many  ciphers 
annexed  as  there  are  orders  in  the  decimal. 

For,  .1  is  equal  to  ^^  ;  .03  is  equal  to  jf  g,  etc. 

3.  Integers  and  decimals  may  form  one  expression. 

For,  both  have  the  same  s,cale,  and  the  value  of  the  unit  denoted  by  a 
figure  in  the  expression  of  each  is  determined  by  its  order  with  refer- 
ence to  the  decimal  point. 

READING   DECIMALS. 

230— Ex.  1.  Read  .308. 

Solution.— .308  =  3  tenths  0  hundredths  8  thousandths ;  or,  .3  =  30 
hundredths  =  300  thousandths,  and  300  thousandths  +  8  thousandths 
=  308  thousandths.     Hence,  .308  may  be  read  SOS  thousandths. 

2.  Read  61.25. 

Solution.  —  G1.25  =  61  ones  and  decimal  .25.  .25  =  2  tenths  5 
hundredths,  or,  since  .2  =  20  hundredths,  20  hundredths  +  5  hundredths 
=  25  hundredths.  Hence,  61.25  may  be  read  01  ones  and  25  hundredths; 
or,  Gl  and.  25  hundredths.     It  may  also  be  read  6125  hundredths. 

3.  Read  .1573;  read  4.59;  read  13.1343;  read  1.0001. 


DECIMAL    FRA  GTIONS. 


105 


231.  Rule  for  Reading  Decimals.— i?e«fZ  the  expression,  if 
a  pure  decimal,  as  an  integer,  giving  it  the  name  of 
its  right-hand  order.    Or, 

Read  the  expression,  if  a  mixed  decimal,  as  a  mixed 
number,  or  as  Us  equivalent  improper  fraction. 


Copy  and  read- 

PROBLEMS. 

1.  .1256. 

4.  .09. 

7. 

.3106. 

2.  .3041. 

3.  .00372. 

5.  .0505. 

6.  .77777. 

8. 
9. 

.60135. 
.000003 

10.  Copy  and  read  as  a  mixed  decimal,  5.19. 

11.  Copy  aud  read  as  an  improper  fraction,  11.5. 
Copy  and  read  as  mixed  numbers — 


12.  6.171. 

13.  10.001. 

14.  91.571 

15.  1.5072. 


16.  45.1145. 

17.  73.00956. 

18.  99.00003. 


19. 


7.000311 


20.  100.000001. 

21.  1005.63004. 

22.  1131.40056. 

23.  134971.023. 


WRITING    DECIMALS. 

232. — Ex.  1.  Write  three  hundred  five  thousandths. 

Solution.  —Three  hundred  five  thousandths  =  305  thousandths  =  30 
hundredths  and  5  thousandths ;  and  30  hundredths  =  3  tenths  and  0 
hundredths.  Hence,  305  thousandths  =  3  tenths  0  hundredths  5  thou- 
sandths =  .305. 

2.  Write  four,  and  fifty-nine  hundredths. 

3.  Write  thirteen,  and  one  thousand  three  hundred  forty- 
two  ten-thousandths. 

233.  Rule  for  Writing  Decimals.— TF/^j^e  the  decimal  as  i 7 /■ 
integers,  and  place  the  decimal  point  so  that  eacJi 
figure  shall  stand  in  its  proper  order,  marking  the 
absence  of  units  of  any  order,  if  necessary ,  by  a  cipher. 


PROBLEMS. 


Write  in  figures — 

1.  Five  hundredths. 

2.  Sixty-seven  hundredths. 


Ans.  .05. 


106  REDUCTION    OF    DECIMALS. 

3.  One  hundred  fifteen  thousandths. 

4.  Thirty-five  ten-thousandths.  Ans.  .0035. 

5.  Seventy,  and  seventy-five  thousandths.      Ans.  70.075. 

6.  Five  hundred  fifty-three  hundred-thousandths. 

7.  Six  hundred,  and  six  and  one-fourth  hundredths. 

Ans.  600.061 

8.  One  thousand  four  hundred  one  millionths. 

9.  One  hundred  thousand,  and  563  hundred-thousandths. 

10.  Fourteen  millions,  and  15003  teu-millionths. 

11.  One  hundred  sixty-three  thousand,  and  one  hundred 
sixty-three  thousandths.  Ans.  163000.163. 

12.  Seven  million,  and  seventy-three  hundred  millionths. 

13.  One  billion  one  million  one,  and  five  hundred  thirty 
thousand  eleven  billionths.         Ans.  1001000001.000530011. 


Write  in  the  decimal  form — 


Ans.  .12. 


ISoWoT-  ^'is- 18.00103. 


18.  T^Wu-  ^^^«-  -2002. 

19.  65^V 

20.  131^0.  Ans.  131.7. 

21.  1000^0  oi)QO(). 


SECTION    XXIV. 
BEDUCTIOJf    OF   DECIMALS. 

C^SE    I. 
Decimals  Reduced  to  a  Common  Denominator. 

234. — Ex.  1.  In  3  tenths  of  a  melon,  hoAV  many  hundredths 
of  a  melon  ? 

2.  In  4  tenths,  how  many  hundredths  ?     In  40  hundredths, 
how  many  thousandths  ? 

3.  Express  4  as  tenths.      .5  as  hundredths.     .25  as  thou- 
sandths.    4.0  as  hundredths. 

4.  Express  5  and  .2  each  in  hundredths.     Express  .4  and 
.15  each  in  thousandths. 


REDUCTION    OF    DECIMALS.  107 

235.  Principle. — Annexing  ciphers  to  a  decimal,  or  removing 
ciphers  from  the  right  of  a  decimal,  does  not  change  the  value 
expressed. 

For  annexing  ciphers  does  not  change  the  order  of  the  other  figures 

with  reference  to  the  decimal  point. 

mL  r         '■n         /r/1/i  ^  50  500 

l^hn.,.5^.o0  =  .500,ov-^  —  =  -j^ 

WRITTEN    EXEItCISES. 

236.— Ex.  1.  Reduce  .9,  .104  and  3.1945  to  equivalent 
decimals  having  the  least  common  denominator. 

g  =  ^QOOO  Solution. — The  smallest  decimal  order  of  any  of 

-I  rx  I  __    i  f)  /  n      the  decimals  is  that  of  ten-thousandths. 

0  -I  n  r  n  Since    annexing    ciphers   to   decimals    does  not 

0.IJ4.O  change  their  values,  we  reduce  .9  to  ten-thousandths 
by  annexing  three  ciphers  and  .104  to  ten-thousandths  by  annexing  one 
cipher,  which  gives  as  the  decimals  required,  .9000,  .1040  and  3.1945. 

2.  Reduce  .006,  7.45  and  .0005  to  a  common  denominator. 

237.  Rule  for  Reduction  of  Decimals  to  a  Common  Denominator.— 

Mahe  the  decimals  have  the  same  number  of  decimal 
orders,  by  annexing  ciphers. 

PJtOBIiEMS. 

1.  Reduce  .15  and  .0016  to  equivalent  decimals  having  a 
common  denominator. 

2.  Change  .5,  .007  and  8  to  similar  decimals. 

Ans.  .500,  .007  and  8.000. 

3.  Reduce  3.4,  .00324  and  .65  to  equivalent  decimals  having 
the  least  common  denominator. 

C^SE  TI, 

Decimals  Reduced  to  Common  Fractions. 

238. — Ex.  1.  In  5  tenths  of  a  melon  there  are  how  many- 
halves  of  a  melon  ? 

2.  How  many  halves  in  y^  ?     In  y^o  •     tWo  ? 

3.  How  many  fourths  in  ^  ?     In  ^^  ?     ^^^  ?     ^^^  ? 

4.  How  many  fifths  in  ,%  ?     In  ^^  ?     ^  ?     ^0^^  ? 

5.  How  many  halves  in  -.5  ?     Fourths  in  .75  ?    Fifths  in  .8  ? 


108  REDUCTION  OF  DECIMALS. 

WniTTEN  EXERCISES. 

239. — Ex.  1.  Reduce  .055  to  an  equivalent  common  frac- 
tion. 

nepr 55 11  SOLUTION. — Omitting  the  decimal   point  and 

1000  200  writing  the  denominator  changes  .055  to  the 
equivalent  common  fraction  xooo>  which,  reduced  to  its  lowest  terms, 
;=  JLi 

^*    2  0  5* 

2.  Reduce  .35  to  an  equivalent  common  fraction. 

240.  Rule  for  Reduction  of  Decimals  to  Common  Fractions.— 
Oinit  the  decimal  point,  ivj%te  the  denominator  under 
the  given  numerator,  and,  if  necessary ,  reduce  the 
fractioji  to  its  lowest  terms. 

PMOBIjEMS. 

Reduce  to  common  fractions  in  their  lowest  terms — 

5.  .08.  A7i8.^. 

6.  1.06. 

7.  .096.  Ans.  ^. 

8.  .006943. 

9.  Reduce  1.06  to  a  mixed  number.  Ans.  1-^. 

10.  Reduce  503.1875  to  a  mixed  number.    Ans.  503^. 

11.  Reduce  .16f  to  an  equivalent  common  fraction, 

1fil-^-J.-lO_l  S0LUTi0N.-.16f  is  equal  to  ^J, 

'^^ 3~  100     ~  100~  S00~~  6        which,  reduced  (Art.  211),  is  \,  the 

fraction  required. 

12.  Reduce  .lO/y  ^^^  ^^  equivalent  common  fraction. 

13.  Reduce  5.3f  to  an  equivalent  mixed  number. 

14.  Reduce  .43-||-  to  an  equivalent  common  fraction. 

A 'it  9      43  0  3 
-^'''*'-     99  00' 

Cj^lSE    III. 


1. 

.0075. 

Ans.  4-3  0- 

2. 

.375. 

3. 

.03125. 

Ans.  -^. 

4. 

.3216. 

^n-s.  m. 

Common  Fractions  Rcdnced  to  Decimals. 

241. — Ex.  1.  In  1  half  of  a  melon,  how  mauy  tenths  of  a 
melon  ?     In  1  fifth  of  a  melon  ? 

2.  How  many  tenths  in  -I?     Hundredths  in  |? 

3.  How  many  hundredths  in  ^?     In  |^?     In  f  ? 


REDUCTION  OF  DECIMALS.  109 

WRITTEN    EXERCISES. 

242. — Ex.  1.  Reduce  f  to  an  equivalent  decimal. 

8)3.000  Solution.— f  is  i  of  3.    3  is  equal  to  30  tenths ;  i  of  30 

a^ry  [~       tenths  is  3  tenths,  with  6  tenths  remaining. 

6  tenths  are  60  hundredtlis;  \  of  60  hundredths  is  7 
Iiundredths,  with  4  hundredths  remaining. 

4  hundredths  are  40  thousandths ;  |  of  40  thousandths  is  5  thousandtlis. 
3  tenths  +  7  Iiundredths  +  5  tliousaudths  =  .375.  Hence,  |  is  equiva- 
lent to  .375. 

2.  Reduce  ^-^  to  an  equivalent  decimal. 

243.  Rule  for  Reduction  of  Common  Fractions  to  Decimals.— JSe- 

duce  the  nuineTator  to  tciitlis,  hundredtlis,  etc.,  hy  an- 
nexing ciphers  ;  divide  the  result  by  the  denominator, 
and  point  off  as  many  orders  for  decimals  in  the 
quotient,  as  there  were  ciphers  anneHced. 

FROHIEMS. 

Reduce  to  decimals — 
1.  2^.  An8.  .36.     I     5.  ||.  Am.  .95. 

6.  i 


2.  f. 

3.  |.  Ans.  2.5. 


7.  -|t|.  Am.  .3216. 

Ans.  .0078125. 


1 

128* 


4.  ^.  An8.  .09375. 

9.  Reduce  3f  to  a  mixed  decimal. 

oS_ ^i^ -^  ly  r  Solution. — 3|^3  and  \.     |is.75;  hence, 

4  4  '  3f  is  3  and  .75,  or  3.75. 

10.  Reduce  47y\  to  a  mixed  decimal.        Ans.  47.1875. 

11.  Reduce  j^  to  a  decimal. 

12.  Reduce  l6o|f  to  a  mixed  decimal.        Am.  100.96. 

13.  Reduce  .151  to  a  pure  decimal. 

14.  Reduce  l.OOf  to  a  mixed  decimal. 

15.  Reduce  503^  to  a  mixed  decimal. 

16.  Reduce  f  to  a  complex  decimal  of  two  orders. 

o  )^  nn  Solution. — |  is  |  of  2.     2  is  equal  to  20  tenths  ;  |  of  20 

— '- tenths  is  6  tenths,  with  2  tenths  remaining. 

.6" 6"-  2  tenths  are  20  hundredths;  i  of  20  hundredths  is  6 

hundredths,  with  2  hundredths  remaining. 
10 


110  BEBVCTION  OF  DECIMALS. 

Here  the  continual  recurring  of  the  same  figure  in  the  result,  with  a 
like  remainder,  shows  that  j  has  no  equivalent  pure  decimal.  In  such 
cases,  when  the  reduction  has  been  carried  to  any  desirable  extent,  we  may 
express  the  remainder  in  the  form  of  a  common  fraction,  and  the  result 
will  be  a  complex  decimal ;  or  we  may  use  the  sign  -|-  to  indicate  the 
incompleteness  of  the  result,  as  .66  +  for  .66|. 

17.  Reduce  ^  to  a  complex  decimal  of  four  orders. 

18.  Reduce  -|^  to  a  decimal  of  four  orders.     Ans.  .1111  -]-. 

19.  Reduce  y\  to  a  decimal  of  four  orders. 

20.  Reduce  ^^  to  a  complex  decimal  of  five  orders. 

21.  Reduce  q-qVo  ^o  ^  decimal  of  six  orders. 

Ans.  .000444+. 

22.  Reduce  -f-^-f  to  a  mixed  decimal  of  five  orders. 


.     TEST    QUESTIONS. 

244. — 1.  "What  is  a  Decimal  Fraction?  Why  is  it  called  decimal? 
How  is  a  decimal,  when  written,  distinguished  from  an  integer?  What 
is  the  denominator  of  a  decimal  ? 

2.  What  do  the  figures  at  the  right  of  the  Decimal  Point  express  ? 
From  what  are  the  names  of  decimal  orders  derived  ?  To  what  do  the 
orders  tenths,  hundredths,  thousandths,  etc.,  correspond  ?  Of  what  does 
a  pure  decimal  consist ?     A  mixed  decimal ?     A  complex  decimal? 

3.  How  many  units  of  any  order  in  a  decimal  are  equal  to  one  of  the 
order  next  higher?  Why  may  integers  and  decimals  form  one  expres- 
sion ?     What  is  the  rule  for  reading  decimals  ?     For  writing  decimals  ? 

4.  How  are  decimals  reduced  to  a  common  denominator?  Why  does 
annexing  a  cipher  to  a  decimal  not  change  its  value  ?  What  is  the  rule 
for  reducing  decimals  to  common  fractions  ? 

5.  What  is  the  Kule  for  reduction  of  common  fractions  to  decimals  ? 
How  do  you  proceed  when  the  common  fraction  has  no  equivalent  pure 
decimal  ? 

6.  What  is  a  Number?  What  are  figures?  Wh;it  is  numeration? 
Notation  ?  What  is  tlie  scale  of  numbers  ?  What  is  tlie  scale  of  the 
ordinary  system?     What  is  termed  the  decimal  system?    (Art.  33.) 

7.  What  is  an  Integer?  A  unit  of  an  integer?  (Arts.  3,  4.)  A  frac- 
tion? A  fractional  unit  ?  The  unit  of  a  fraction  ?  (Arts.  141,  143.)  A 
decimal  fraction  ?  How  is  a  common  fraction  expressed  by  figures  ?  How 
ia  a  decimal  expressed  by  figures  ?  How  do  integers  and  decimals  cor- 
respond in  expression  ?    (Art.  228.) 


REDUCTION  OF  DECIMALS.  Ill 

SECTION    XXV. 

ADDITIOJf  AJfD  SUBTRACTIOJ^  OF  DECIMALS. 

245. — Ex.  1.   How  many  tenths  are  5  tenths  and  4  tenths? 

2.  How  many  tenths  are  -^  and  y\  ?     .4  and  .3  ? 

3.  How  many  hundredths  are  y^^^  and  y^-j^  ?     .43  and  .8  ? 

4.  How  many  hundredths  are  -^-^  less  y^-^  ?     .51  less  .8  ? 

246.  Principle. — Decimals  which  are  similar  may  he  added 
or  subtracted  like  integers. 

WRITTEN  EXERCISES. 

247.— Ex.  1.  Add  13.634,  35.423  and  8.56. 

13.6 3 A  Solution. — Writing  the  numbers  so  that  all  the  figures 

op:  /(j)o       of  the  same  order  stand  in  the  same  column,  and  adding 
as  in  addition  of  integers,  gives  57.617,  the  sum  required. 
The  8.56  is  made  similar  to  the  other  decimals  by  an- 


8.560 


57 .6 17       nexing  a  cipher. 

2.  From  963.75  subtract  585.125. 

963.750  Solution. — Writing  the  numbers  so  that  figures  of  the 

/- o/r   i pp:      same  order  stand  in  the  same  column,  and  subtracting  as 

in  subtraction  of  integers,  gives  378.625,  the  difierence  re- 

378.625       qnired. 

The  cipher  annexed  to  the  minuend  is  usually  understood,  and  the 
subtraction  performed  in  the  same  manner  as  if  it  were  written. 

3.  What  is  the  sum  of  145.07,  3.476  and  11.05? 

4.  What  is  the  difference  between  56.77  and  7.899  ? 

248.  Rule  for  Addition  and  Subtraction  of  Decimals.  —  Write 
the  Tiumhers  so  that  figures  of  the  same  order  shall  he 
iiv'the  same  coluimi. 

Add  or  subtract  in  the  same  Tnanner  as  if  the  num- 
hers  were  integers,  and  place  the  decimal  point  at  the 
left  of  the  order  of  tenths  in  the  result. 


112  REDUCTION  OF  DECIMALS. 

FltOBXEMS. 

1.  What  is  the  sum  of  .89,  .269, 15.2  and  .2?    Ans.  16.559. 

2.  What  is  the  sum  of  11.35,  19,  3.41  and  100.678? 

3.  What  is  the  difference  between  .9173  and  .2138  ? 

Ans.  .7035. 

4.  AVhat  is  the  difference  between  407  and  91.713? 

Ans.  315.287. 

5.  Required  the  value  of  270.2  less  75.4075. 

6.  450  +  376.004  +  1.08  +  .76  +  .05  =  what  number? 

Ans.  827.894. 

7.  .001  — .00099  =  what  number?  Ans.  .00001. 

8.  100  —  .10101  =  what  number  ?  Ans.  99.89899. 

9.  What  is  the  sum  of  ten  thousand  one  hundred  one  thou- 
sandths, ninety-nine,  and  eighty-nine  thousand  eight  hundred 
ninety-nine  hundred  thousandths  ? 

10.  What  is  the  sum  of  98.75  miles,  100.3655  miles  and 
15.7875  miles? 

11.  If  41.674  cubic  feet  of  oak  wood,  or  64.693  cubic  feet 
of  white  pine  wood,  will  Aveigh  a  ton,  how  many  more  cubic 
feet  are  there  in  a  ton  of  pine  than  in  a  ton  of  oak  ? 

Am.  23.019  cubic  feet. 

12.  Add  seventy -five  hundredths;  eight,  and  sixty-seven 
hundredths ;  seven,  and  three  hundred  fifty-five  thousandths ; 
and  thirty-one,  and  seven  hundred  thirty-five  thousandths. 

Am.  48.51. 

13.  From  three  millions  three,  take  three,  and  three  mil- 
lionths. 

14.  In  one  field  there  are  31.175  acres ;  in  another,  9.1825 
acres ;  and  in  a  third,  25.75  acres.  How  many  acres  in  the 
three  ? 

15.  What  part  of  1  must  be  added  to  9.999999  to  make  10? 

Ans.  One  millionth  part. 

16.  From  the  sum  of  one  and  five  hundredths,  eighteen  and 
thirty-five  ten-thousandths,  and  four  hundred  four  millionths, 
subtract  eleven  and  ninety-nine  ten-millionths. 

Am.  8.0538941. 


MULTIPLICATION  OF  DECIMALS.  113 

section  xxvi. 
multiplicjltiojY  of  decimals. 

249. — Ex.  1.  How  many  tenths  of  a  dollar  are  3  times  3 
tenths  of  a  dollar  ? 

2.  How  many  tenths  are  3  times  3%  ?     4  times  y^  ? 

3.  How  many  hundredths  of  a  dollar  are  7  times  6  hun- 
dredths of  a  dollar  ? 

4.  How  many  hundredths  are  7  times  j^  ?     6  times  .08  ? 

5.  How  many  hundredths  in  j^  X  yo  '•     -6  X  .4  ? 

6.  How  many  thousandths  in  jwo  X  8  ?     y^^^  X  1%  ? 

7.  How  many  ones  are  10  times  y\  ?     10  times  .6  ? 

8.  How  many  tenths  are  10  times  yf  ^  ?     10  times  .05  ? 

9.  How  many  hundredths  are  9  times  yto?  How  many 
tenths  are  3  times  .9  ? 

250.  Principles. — 1.  Each  removal  of  the  decimal  point  one 
order  to  the  right  makes  the  value  of  an  expr&ssion  ten-fold. 

For,  by  the  removal,  each  figure  is  made  to  express  units  of  the  next 
higher  order. 

Thus,  16.4  =  sixteen  and  four  tenths ;  and  164.  =  one  hundred  sixty- 
four,  or  ten  times  16.4. 

2.  Each  removal  of  the  decimal  point  one  order  to  the  left  makes 
the  value  of  an  expression  one-tenth  as  large  as  before. 

For,  by  the  removal,  each  figure  is  made  to  express  units  of  the  next 
lower  order. 

Thus,  164.  =  one  hundred  sixty-four ;  and  16.4  =  sixteen  and  four- 
tenths,  which  is  one  tenth  as  much  as  164. 

WRITTJEK    EXERCISES. 

251.— Ex.  1.  Multiply  .49  by  6. 
.A9  Solution. — 6  times  9  hundredths  are  54  hundredths,  or  5 


6 


tenths  and  4  hundredths. 

Write  4  for  the  hundredths  of  the  result,  and  reserve  the  5 
^.94-      tenths.     6  times  4  tenths  are  24  tenths  ;  24  tenths  and  5  tenths 

are  29  tenths,  or  2  ones  and  9  tenths.     Write  2  ones  9  tenths  in 
the  result,  which  gives  2.94,  the  product  required. 


114  MULTIPLICATION    OF   DECIMALS. 

2.  Multiply  7.6  by  .06. 

7"  Q  Solution. — .00  is  the  same  as  y^^j  of  6  ;  hence,  .06  times  7.6 

/^^P       is  the  same  as  j^^  of  6  times  7.6. 

— 6  times  7.6  are  45.6,  and  j^u  of  6  times  7.6  is  j^q  of  45.6, 

.4-56       which,  found  by  removing  the  decimal  point  two  orders  to  the 
left,  is  .456.     Hence,  7.6  multiplied  by  .06  is  .456. 

Or,  the  solution  may  be  explained  as  follows : 

6  hundredtlis  times  6  tenths  is  36  thousandths,  or  3  hundredths  and  G 
thousandths.  Write  6  in  the  thousandths'  order  in  the  result,  and  re- 
serve tlie  3  hundredths. 

6  hundredtlis  times  7  is  42  hundredths  ;  42  hundredths  and  3  hun- 
dredtlis are  45  liundredtlis,  or  4  tenths  and  5  hundredths,  which  we  write 
in  the  result,  and  have,  as  before,  .456. 

By  the  first  explanation  it  appears  that  in  decimals,  as  in 
common  fractions  (Art.  195), 

Multiplying  by  a  fraction  is  the  same  as  multiplying  by  its 
numerator  and  dividing  the  result  by  its  denominator. 

By  that  process,  when  the  factors  are  decimals. 

The  number  of  decimal  orders  in  the  jiroduct  is  made  as  many 
as  there  are  in  both  factors. 

3.  Multiply  1.704  by  .35.  .    Ans.  .5964. 

4.  Multiply  .0051  by  51.  Ans.  .2601. 

252.  Rules  for  Multiplication  of  Decimals.— i.  Multiply  as  in 
integers,  and  place  the  decimal  point  in  the  pro- 
duct, so  that  it  shall  have  as  many  decimal  orders 
as  are  contained  in  both  factors.    Or, 

2.  If  the  multiplier  is  a  decimal,  multiply  hy  its 
numerator  and  divide  hy  its  denominator. 

PROBL  EMS. 

Multiply — 


1.  .125  by  .025.   Ans.  .003125. 

2.  8.25  by  4.5. 

3.  958  by  .34.        Ans.  325.72. 

4.  500.83  by  121. 

Ans.  60600.43. 


6.  4.6337  by  100.  ^»s.  463.37. 

7.  4.6337  by  1000. 

8.  3.007  by '.36.  Ans.  1.08252. 

9.  .285  by  .003. 

10.  3.84062  by  70000. 


5.  1.007  by  .0041.  I  Ans.  268843.4. 


DIVISION    OF   DECIMALS.  115 

11.  What  is  the  product  of  9.688  by  .2|?       Ans.  2.6642. 

12.  What  will  56  pounds  of  coffee  cost,  at  .37|-  of  a  dollar 
per  pound  ? 

13.  If  a  box  must  have  a  capacity  of  24.958  cubic  feet  to 
contain  a  ton  of  anthracite  coal,  what  must  be  the  capacity  of 
a  box  that  will  contain  4.5  tons  ? 

14.  What  is  the  weight  of  128  cubic  feet  of  common  soil; 
if  the  weight  of  a  cubic  foot  is  137.125  pounds  ? 

Ans.  17552  pounds. 

SECTION    XXVII. 

DIVISION    OF   DECIMALS. 
253. — Ex.  1.  In  9  tenths  of  a  dollar  hoAV  many  times  3  tenths? 

2.  How  many  times  3  tenths  is  -^^  ?     Is  .9  ? 

3.  In  42  hundredths  of  a  dollar  how  many  times  6  hun- 
dredths of  a  dollar  ? 

4.  How  many  times  is  .8  contained  in  .48  ?     .3  in  .06  ? 

5.  What  is  6  ^  10  ?    .6  -  10  ?     .6  ^.6  ?     .06  ^  .6  ? 

WniTTEN  EXERCISES. 

254— Ex.  1.  Divide  1.345  by  5. 

5)1.34-5  Solution.— 1.345  is  1345  thousandths;  \  of  1345  thoii- 

.269      S'indths  is  269  thousandths,  or  .269. 

2.  Divide  1.46  by  .25. 

25)146.00(5.84         Solution.-.25  is  the  same  as  fi^,  or  ^^^  X  25. 
19^  First,  divide  1.46  by  j^q-,  by  multiplying  by 

■ 100,  which  is  done  by  removing  the  decimal 

'^-''^  point  in  the  dividend  two  orders  to  the  right, 
200  making  the  dividend  146.    146  divided  by  25  is 
inn  5,  with  21   ones  remaining.     21  ones  are  210 
.                           tenths,  which   divided  by  25  is  8  tenths,  with 
10  tenths  remaining.      10  tenths  are  100  hun- 
dredths,   which,   divided    by    25,   is    4    hun- 
dredths.    Hence,  1.46  divided  by  .25  =  5.84.     Or, 

Multiplying  both  the  divisor  and  dividend  by  100,  the  denominator  of 
the  divisor,  making  the  divisor  a  whole  number,  and  then  dividing,  we 
have  the  quotient  5.84,  as  before. 


116  DIVISION  OF  DECIMALS. 

By  the  first  explanation  it  appears  that  in  decimals,  as  in 
common  fractions  (Art.  206), 

Dividing  by  a  fraction  is  performed  by  multiplying  by  its  de- 
nominator and  dividing  the  residt  by  its  numerator. 

By  that  process  in  the  division  of  decimals, 

The  number  of  decimal  orders  in  the  quotient  is  made  as  many 
as  there  are  in  the  dividend,  less  the  number  in  the  divisor. 

3.  Divide  .5964  by  .35.  Ans.  1.704. 

4.  Divide  .2601  by  51.  Ans.  .0051. 

255.  Rules  for  Division  of  Decimals.— i.  If  the  divisor  is  an 
ijvteger,  divide  as  in  integers,  and  point  off  as  many 
decimal  oi^ders  in  the  quotient  as  there  are  decimal 
orders  in  the  dividend. 

'2.  If  the  divisor  is  a  decimal,  mahe  it  an  integer  by 
lyvoving  the  decimal  point  to  the  right,  and  move  the 
decii^val  point  in  the  dividend  as  many  orders  to  the 
right,  and  then  divide.     Or, 

3.  If  the  divisor  is  a  decimal,  multiply  the  dividend 
by  its  denominator  and  divide  the  i^esult  by  the  nu- 
merator. 

When  the  divisor  is  10,  100,  etc.,  the  division  may  be  performed  sim- 
ply by  removing  the  decimal  point  in  the  dividend  as  many  orders  to  the 
left  as  there  are  ciphers  in  the  divisor. 

PROBLE31S. 


Divide — 

1.  .456  by  .06.  Ans.  7.6. 

2.  463.37  by  100. 

3.  1.606  by  44.        Ans.  .0365. 

4.  1.08252  by  .36. 

5.  825.72  by  958.        Ans.  .34. 


6.  463.37  by  1000. 

7.  21  by  56.  Ans.  .375. 

8.  .1606  by  44.    Ans.  .00365. 

9.  6  by  .006.  Ans.  1000. 
10.  172.8  by  .0144. 


11.  What  is  the  value  of  21.17--  .0073? 

12.  What  is  the  value  of  .015625  -^  25  ?     Ans,  .000625. 

13.  What  is  the  value  of  2.15565  h-  1.05  ? 

14.  What  is  the  quotient  of  3.672  by  .81  ?    Ans.  4.533+. 


DIVISION  OF  DECIMALS.  117 

When,  as  in  tlie  last  problem,  the  division  will  not  terminate,  the  sign 
+  may  be  annexed,  to  indicate  that  the  quotient  is  not  complete.  The 
quotient  is  then  called  an  approximate  quotient. 

15.  What  is  the  approximate  quotient  of  45.5  divided  by 
2100  ?  Ans.  .0216+. 

16.  If  you  should  travel  787.5  miles  in  210  hours,  at  what 
rate  per  hour  Avould  you  travel  ? 

17.  If  .0001  is  a  dividend  and  1.25  a  divisor,  what  is  the 
quotient  ?  Ans.  .00008. 

18.  Divide  three  thousand  one  hundred  tAventy-five  mil- 
lionths,  by  one  hundred  twenty-five  thousandths. 

19.  If  375  bushels  of  potatoes  be  Avorth  as  much  as  7.5  tons 
of  hay,  hoAV  many  bushels  are  Avorth  as  much  as  1  ton? 

Ans.  50. 

20.  A  tract  of  land  containing  125.4  acres  was  sold  for 
7586.7  dollars  ;  Avhat  was  the  price  per  acre  ? 

21.  How  many  casks,  each  containing  31.5  gallons,  can  be 
filled  from  a  vat  containing  368.25  gallons  ? 

Ans.  11  ;  and  21.75  gallons  remain. 

22.  If  16.284  cubic  feet  of  fire-bricks  weigh  one  ton,  how 
many  loads,  of  a  ton  each,  Avill  a  pile  of  such  bricks,  contain- 
ing 333.822  cubic  feet,  make? 

Ans.  20  ;  and  8.142  cubic  feet  remain. 


TEST   QUESTIONS. 

256. — 1.  What  kind  of  decimals  can  be  added  or  subtracted? 
What  is  the  rule  for  addition  or  subtraction  of  decimals  ?  What  does 
the  decimal  point  in  the  result  of  addition  or  subtraction  mark  ? 

2.  In  \Vhat  simple  way  may  the  value  of  a  decimal  expression  be 
MULTIPLIED  by  10  ?  Why  does  the  removal  of  the  decimal  point  in  an 
expression  one  order  to  the  right  make  the  value  expressed  tenfold  ? 
What  are  the  rules  for  multiplication  of  decimals  ? 

3.  In  what  simple  way  may  the  value  of  a  decimal  expression  be 
biviDED  by  10  ?  Why  does  the  removal  of  the  decimal  point  in  an  ex- 
pression, one  order  to  the  left,  make  the  value  expressed  one  tenth  as 
large  ?  What  are  the  rules  for  division  of  decimals  ?  When  is  a  quo- 
tirtit  called  an  approximate  quotient? 


lis 


UNITED    STATES    MONEY. 


UJ^ITED     STATES 
MOXEY. 

257.  Money  is  a  measure 
of  value  used  as  a  medium 
of  trade. 

258.  Coin,  or  Specie,  is  metal  stamped, 
and  authorized  by  government  to  be  used 
as  money. 

259.  Paper  Money  consists   of    Notes, 
issued  by  banks  and  by  the  Treasury  of 
the  United  States,  as  substitutes  for  coin. 
Treasury  Notes  of  less  face-value  than 
are  called  Fractional  Currencv. 

200.  Currency  is  the  Coin  and  Notes 
in  circulation  as  money. 

Bidiion  is  uncoined  gold  or  silver.  An  Alloy 
is  a  baser  metal  mixed  with  a  finer. 

A  Token  is  a  coin  whose  intrinsic  value  is  less  than  that  assigned  it  by 
law. 

261.  United  States  Money   is   the   currency  of  the  United 
States. 


UXITED    STATES    MONET.  119 

TABLE. 
10  mills  (in.  J  are  1  cent . .  .ct.  or  c. 

10  cents  "     1  dime d. 

10  diines,  or  100  cents,    "     1  dollar $. 

262.  The  Coins  of  the  United  States  are  made  of  gokl, 
silver,  nickel  and  bronze. 

The  Gold  Coins  are  the  fifty-dollar  piece,  double  eagle  or 
twenty-dollar  piece,  eagle  or  ten-dollar  piece,  half-eagle,  quarter- 
eagle,  three-dollar  piece  and  dollar. 

The  Silver  Coins  are  the  dollar,  half-dollar,  quarter-dollar, 
dime,  half-dime  and  three-cent  piece. 

The  Nickel  Coins  are  the  five-cent  piece  and  three-cent  piece. 

The  Bronze  Coins  are  the  two-cent  piece  and  cent. 

The  gold  coins  are  made  of  9  parts  of  pure  gold  and  1  part  of  an 
alloy  consisting  of  silver  and  copper.  The  silver  coins  are  made  of  9 
parts  of  pure  silver  and  1  part  of  copper.  The  nickel  coins  are  made 
of  75  parts  of  copper  and  25  parts  of  nickel.  The  bronze  coins  are 
made  of  95  parts  of  copper  and  5  parts  of  zinc  and  tin. 

263.  Canada  Money,  or  the  money  of  the  Dominion  of  Canada, 
consists,  like  United  States  money,  of  dollars  and  cents.  Of 
this  money,  100  cents  are  1  dollar. 

The  Canada  coins  are  the  twenty-cent,  ten-cent  and  five-cent  pieces, 
made  of  silver,  and  the  cent,  made  of  bronze. 

264.  The  Dollar  is  the  principal  unit  of  United  States  money. 
Dimes,  cents  and  mills  may  be  written  respectively  as  tenths, 

hundredths  and  thousandths  of  dollars,  and,  when  decimally 
expressed,  may  be  separated  from  dollars  by  the  decimal  jioint. 

265.  Dimes,  or  tens  of  cents,  are  commonly  regarded  as  a 
number  of  cents. 

Thus,  15  dollars  3  dimes  6  cents  7  mills  are  written  $15,367,  and  read 
fifteen  dollars  thirty-six  cents  seven  mills. 

266.  Any  decimal  of  a  dollar  less  than  a  cent  may  be  read 
as  a  decimal  of  a  cent. 

Thus,  $42.5025  may  be  read  forty-two  dollars  fifty  and  twenty-five  hun- 
dredths cents. 


120  USITED   STATES  MONEY. 

WRITTES   EXERCISES. 

267.  Write  and  read — 

1.  $2.45;   $17.17;   $43.47;   $.95;   $60. 

2.  $.05;    $.005;   $.555;   $.606;   $6.06. 

3.  $.0556;   $.7585;   $.7008;   $70.08756. 

4.  $300.30;  $45,303;   $90,909;   $40.0025. 

Write— 

5.  Fifty  cents ;  five  cents  five  mills. 

6.  Two  dollars  twenty-five  cents ;  thirty  dollars  four  cents. 

7.  11  dollars  37  cents;  213  dollars  73  cents. 

8.  7  mills ;  31  cents  5  mills ;  14  cents  6  mills. 

9.  3  dollars  -^^ ;  15  cents  5  mills ;  7  dollars  5  mills. 

10.  38  and  5  tenths  cents ;  17  dollars  17  cents ;  7  tenths  mills. 

11.  3  and  3  tenths  cents ;  15  and  75  hundredths  dollars. 

12.  63  dollars  63  cents ;  6  and  5  tenths  mills. 

13.  600  dollars  6  and  6  tenths  cents. 


SECTION    XXIX. 
REDUCTIOJ^  OF   UjYITED  STATES  MOJ^EY. 

268. — Ex.  1.  How  many  mills  are  9  cents?     Are  11  cents? 

2.  How  many  cents  are  $3  ?    Are  $5  ?     Are  $6  ? 

3.  In  $1.15,  how  many  cents?     In  $2.25 ? 

4.  In  90  mills,  how  many  cents?     In  110  mills? 

5.  How  many  mills  are  5  dimes  ?     Are  $5  ? 

6.  How  many  dollars  are  300  cents ?     Are  115  cents?     Are 
500  cents  ?     Are  5000  mills  ? 

DEFINITIONS. 

269.  Denomination  is  the  name  of  the  unit  expressing  a 
number. 

Of  two  denominations,  the  higher  is  that  which  expresses  the  greater 
value,  and  the  loicer  is  that  wliich  expresses  tlie  less  value. 

270.  Reduction  is  the  j^rocess  of  changing  a  number  to  an 
equivalent  number  of  a  different  denomination. 


UNITED  STATES  MONEY.  121 

}r KITTEN  EXERCISES. 

271. — Ex.  1.  Reduce  56  ceuts  to  mills. 

^/' V  in--^pn  Solution. — Since  1  cent  is  10  mills,  56  cents 

must  be  56  times  10  mills,  or  560  mills. 

2.  Reduce  $43  to  cents. 

/o^  -t nn  —  /<onn  Solution. — Sincfe  $1  is  100  cents,  $43  must 

be  4o  times  100  cents,  or  4ti00  cents. 

3.  Reduce  $25.46  to  cents. 

nr    in^/ir\n      ^cn-  SOLUTION.— Since  $1  is  100  cents,  $25.46 

i!^o.4oX  luO—zoAb  ^,    oc  lo  ^-        iaa       .       »-L 

^         must  be  25.46  times  100  cents,  or  2o46  cents. 

4.  Reduce  $43  to  mills. 

/'?V  mnn  --  f^OOn  Solution.— since  $1  is  1000  mills,  $43 

5.  Reduce  560  mills  to  cents. 

f'Pf\_^  -J  r\ rp  Solution. — Since  10  mills  are  1  cent,  560  mills 

must  be  as  many  cents  as  10  mills  are  cont*iined 
times  in  560  mills,  which  are  56  times.     Hence,  560  mills  are  56  cents. 

6.  Reduce  2546  cents  to  dollars. 

nr  1^       -//^^      r>r    in  SOLUTION. — Since  100  cents  are  $1,  2546 

254-6^  100^ 25. 4.b         ,        ,,  ,  „         /'     , 

cents  must  be  as  many  dollars  as  100  cents 
are  contained  times  in  2546  cents,  which  are  25.46  times.  Hence,  2546 
cents  are  $25.46. 

7.  Reduce  43000  mills  to  dollars. 

f^nnn—  mnn—  /<?         Solution.— Since  1000  mills  are  $1,  43000 
'^'  '^'        mills  must  he  as  many  dollars  as  1000  mills 

are  contained  times  in  43000  mills,  which  are  43  times.     Hence,  43000 
mills  are  $43. 

8.  How  many  mills  are  75  cents  ? 

9.  How  many  cents  are  750  mills  ? 

10.  How  many  mills  are  $17? 

11.  How  many  dollars  are  17000  mills? 

272.  Rules  for  Reduction  of  United  States  Money.— i.  To  reduce, 
cejits  to  mills,  multiply  hy  10 ;  dollars  to  cents,  mul- 
tiply by  100 ;  and  dollars  to  mills,  multiply  hy  1000 ; 
and  give  the  sign  of  the  denomination  as  required. 


122  VNITED  STATES  MONEY. 

2.  To  reduce  dollars  and  cents  to  cents,  or  dollars, 
cents  and  mills  to  uvills,  simply  remove  the  dollar-sign 
ai%d  the  decimal  point. 

3.  To  reduce  mills  to  cents,  divide  hy  10;  cents  to 
dollars,  divide  by  100 ;  ajid  mills  to  dollars,  divide 
hy  1000 ;  and  give  the  sign  of  the  denomiiiation  as 
requii^ed. 


PROBLEMS. 

Reduce  to  mills —                            Reduce  to  cents — 

1. 

62^  cents.        Ans.  625m. 

4.  875  mills.      Ans.  87|  cts. 

2. 

$13. 

5.  $93.50. 

3. 

$6,755. 

6.  6790  mills.     Ans.  679  cts 

7.  How  many  cents  are  $14.50?                        Ans.  1450. 

8.  How  many  dollars  are  1560  cents?              Ans.  15.60. 

9.  How  many  dollars  are  14444  mills  ? 

10.  Reduce  2089  mills  to  dollars. 

11.  Reduce  45395  mills  to  dollars.                 Ans.  $45,395. 

SECTION  XXX. 

COMPUTATlOJfS  IJY  UNITED  STATES  MOJ{EY. 

273.  Computations  in  United  States  Money,  since  in  it  the 
dollar  is  regarded  as  the  unit  in  a  decimal  system  of  notation, 
are  performed  as  in  integers  and  decimals. 

In  final  results  of  computations  such  parts  of  a  cent  as  are 
halves,  fourths  or  eighths  are  generally  expressed  as  common 
fractions. 

WRITTEN  EXERCISES. 

274— Ex.  1.  What  is  the  sum  of  $5.87i  $31,  $7.50  and 
$4.37.^?  "      ^m  $48.75. 

2.  Add  $13,625,  $92.50,  $60  and  $31^. 

3.  I  paid  $5.75  for  a  hat,  $7  for  a  pair  of  boots,  $1.25  for  a 
handkerchief,  and  $43.75  for  a  suit  of  clothes.  How  nuich  did 
the  whole  cost  me  ?  Ans,  $57J5. 


VXITED    STATES   MONEY.  123 

4.  What  is  the  sum  of  $100.50,  $79.62|,  $1.67  and  $19.06^? 

5.  A  merchant  has  on  deposit  in  one  bank  $1634.55 ;  in  an- 
other, $37007  ;  in  a  third,  $3106.75 ;  and  has  on  hand  $563.79. 
How  much  money  has  he  in  all  ?  Ans.  $42312.09. 

6.  A  gentleman  has  a  house  in  the  city  worth  $21360,  which 
is  $9836.75  more  than  his  farm  cost  him.  What  did  his  farm 
cost  him  ? 

7.  I  bought  goods  to  the  amount  of  $33767.50  on  credit, 
and  to  the  amount  of  $9000  for  cash.  How  much  does  the 
former  amount  exceed  the  latter  ? 

8.  If  a  man's  property  is  valued  at  $25360.50,  and  his  debts 
are  $13675.875,  what  is  he  worth?  Ans.  $11684.62^. 

9.  Wilson  has  dry-goods  worth  $931.45,  groceries  worth 
$833. 97|,  hardware  worth  $363,311  a  store  worth  $9000,  and 
land  worth  $1000.75.     How  much  is  he  worth  ? 

10.  If  you  should  save  31  cents  a  day  for  313  working  days, 
what  would  be  the  amount  saved  ?  Ans.  $97.03. 

11.  A  father  divided  equally  among  his  9  children  $28369.80. 
What  Avas  the  share  of  each  ? 

12.  How  much  must  you  save  a  day  to  lay  by  $97.03  in  313 
days  ? 

13.  How  much  must  be  paid  for  transporting  5673  bushels 
of  wheat  at  41  cents  a  bushel  ?  Ans.  $255,281 

14.  If  a  man  spend  12  cents  a  day  for  cigars,  and  25  cents 
for  drink,  how  much  will  he  spend  in  one  year,  or  365  days  ? 

15.  How  much  is  the  rent  of  a  farm  of  2451  acres  at  $3.40 
per  acre  ?  Ans.  $834.70. 

16.  If  it  requires  $7.50  to  give  a  certain  number  of  boys 
12}  cents  each,  what  is  the  number  of  boys? 

17.  A  drover  who  had  $276.25,  bought  5  cows  of  equal 
value,  which  took  all  the  money  he  had,  except  $50.  What 
Avas  the  cost  of  each  cow  ?  Ans.  $45.25. 

18.  How  many  bushels  of  wheat,  at  a  freight  of  4|-  cents  a 
bushel,  must  be  transported  for  $255.28|-  ? 

19.  I  bought  37  hogsheads  of  molasses,  but  the  price  falling, 
I  was  obliged  to  sell  it  at  a  loss  of  $173.53.  What  was  the 
loss  on  each  hogshead  ?  Ans.    $4.69. 


124  UNITED  STATES  MONEY. 

SECTION    XXXI. 

BUSIJfESS  METHODS. 

275. — Ex.  1.  How  much  must  be  paid  for  8  barrels  of  flour, 
at  $9  per  barrel  ? 

2.  When  $72  is  paid  for  8  barrels  of  flour,  how  much  is 
paid  for  1  barrel  ? 

3.  How  many  barrels  of  flour,  at  $9  a  barrel,  can  be  bought 
forS72? 

4.  What  part  of  a  dollar  is  12^  cents?  Is  25  cents?  Is  50 
cents  ? 

5.  At  12^  cents  per  pound,  how  many  pounds  of  sugar  can 
be  bought  for  a  dollar  ? 

6.  At  25  cents  a  yard,  how  many  yards  of  cambric  can  be 
bought  for  a  dollar  ? 

7.  At  12^  cents  per  pound,  how  much  must  be  paid  for  8 
pounds  of  sugar  ?     For  24  pounds  ? 

8.  At  25  cents  a  yard,  how  much  must  be  paid  for  4  yards 
of  cambric  ?     For  40  yards  ? 

9.  At  37^  cents,  or  $f  per  yard,  how  much  must  be  paid  for 
IG  yards  of  cloth?     For  48  yards? 

10.  At  75  cents,  or  $f  per  pound,  what  will  8  pounds  of  tea 
cost  ?     24  pounds  ? 

DEFINITIONS. 

276.  Price  is  the  money-value  assigned  to  the  measuring  unit 
of  any  commodity. 

277.  Quantity,  in  the  sale  of  property,  is  the  amount  of  any 
commodity,  and  is  expressed  by  the  number  of  times  it  con- 
tains the  measuring  unit. 

278.  Cost  is  the  value  assigned  to  an  entire  quantity. 

279.  An  Aliquot  Part  of  a  number  is  an  exact  half,  third, 
fourth,  etc.,  or  any  exact  fractional  part  of  that  number. 
Hence,  the  aliquot  parts  of  a  number  are  found  by  succes- 
sive divisions  of  the  number  by  2,  3,  4,  5,  etc. 


UNITED   STATES  MONEY. 


125 


In  Business,  frequent  use  is  made  of  the  convenient  aliquot 
parts  of  a  dollar  given  in  the  following 


5  cts.  =  -^  of  $1. 
10  cts.  ^Y^  of  $1 


TABLE. 

6,  cts. 

4 

=iof$i. 

8g  cts. 

=  j^of$l. 

l^l  cts. 

-jof$l. 

lel  cts. 

=iof$i. 

33 j  cts. 

=iof$i. 

CAS 

E    I. 

20  cts.  =  j  of  $1. 
2o  cts.  =  70/  ^1^ 
50  cts.  =^  of  Si- 


To  Find  the  Cost  of  a  Given  Quantity  when  the  Price  is  an  Aliquot 
Part  of  a  Dollar. 

280.— Ex.  1.  At  $.25  a  bushel,  what  will  856  bushels  of 
apples  cost  ? 

A)85G  ^=  No.  bushels.        SoLtixiON. — At  $1  a  bushel,  856  bushels  will 
'oT/       AT     J  77  cost  $856;  and  at  $.25,  or  1^  of  a  dollar,  a  bushel, 

214  =  No.  dollars,     g^g  bughej^  ^jh  cost  \  of  $856,  or  $214. 

2.  At  $.12|-  a  pound,  what  will  960  pounds  of  sugar  cost? 

281.  Rule  for  Finding  the  Cost  when  the  given  Price  is  an  Aliquot 
Part  of  a  DoWar.—Mj^st  find  the  cost  at  ^1,  and  then  tahe 
the  aliquot  part  of  this  amount. 


PltOBI^EMS. 

1.  How  innch  Avill    796   bushels  of  oats  cost,  at  $.50  per 
bushel  ? 

2.  At  $.33^  per  yard,  how  much  must  be  paid  for  6  pieces 
of  dress  goods,  each  piece  containing  31  yards  ?      Ans.  $62. 

3.  How  much  must  be  paid  for  1986  pounds  of  granulated 
sugar,  at  $.16f  a  pound? 

4.  What  must  be  paid  for  9684  pounds  of  cheese,  at  $.12^ 
per  pound?  ^«s.  $1210.50. 

11  * 


126  V SITED  STATES  MONEY. 

C^SE    II. 

To  Find  the  Cost  when  the  Price  of  100  or  1000,  and  the 
Quantity,  are  given. 

282.— Ex.  1.  At  $225  per  hundred,  what  will  950  choice 
pear  trees  cost  ? 

$225 

'^•^^  Solution.— 950  is  9.50  hundreds.     If  1  luindred  cost 

11250      $225,   9.50   hundreds   must    cost    9.50   times   $225,   or 
2025  $2137.50. 

$2137.50 

2.  At  S14  a  thousand,  what  will  5545  bricks  cost? 

5M5 

^■^  Solution. — 5545  is  5.545  thousands.     At  $14  a  thou- 

22180      sand,  5.545  thousands  will  cost  as  many  dollars  as  the  prod- 
5545  uct  of  5.545  by  14,  or  $77.63. 


$77,630 


3.  How  much  must  be  paid  for  11125  hoop-poles,  at  $40  per 
thousand?  Ans.  $445. 


28^.  Rule  for  Finding  the  Cost  when  the  Price  of  100  or  1000  is 
%\yen.— Point  off  the  ninnbcr  expressing  the  quantity 
into  hundreds  or  thousands,  as  inay  be  required,  and 
find  the  product  of  the  price  and  this  numher. 

PROBLEMS. 

1.  What  will  be  the  cost  of  11750  feet  of  pine  boards,  at  $55 
per  thousand  feet?  Ans.  $646.25. 

2.  What  will  5635  melons  cost,  at  $14  per  hundred  ? 

3.  What  is  the  freight  on  19362  pounds  of  merchandise,  at 
$.45  per  thousand?  A)is.  $8.71  +. 

4.  What  will  7536  fish  cost,  at  $3.13  per  hundred? 

5.  How  mucli  will  15500  laths  cost,  at  $.90  per  hundred, 
and  17960  feet  of  timber,  at  $35  per  thousand  feet? 


UNITED   STATES  MONEY.  W, 


CJ^mSi    III. 


To  Find  the  Cost  when  the  Price  of  a  Ton  of  2000  pounds,  and 
the  Quantity,  are  given. 

284.— Ex.  1.  What  Avill  4800  pounds  of  coal  cost,  at  $7.55 
per  ton  ? 

2)4.800  $7.55  Solution. — 4800  pounds  are  4.800  tliou- 

sand  pounds. 

^.Jf-UU  /i.Ji-  Since  2000  pounds  are  1  ton,  there  must 

3020      ^^  ^^'^'^  as  many  tons  as  there  are  thousand 
7^7^)  pounds,  or  2.4  tons. 

At  $7.55  per  ton,  2.4  tons  will  cost  2.4 


.  120      times  $7.55,  or  $18.12. 

2.  What  will  9756  pounds  of  hay  cost,  at  $25  per  ton  ? 

Ans.  $121.95. 

3.  What  will  11560  pounds  of  coal  cost,  at  $9  per  ton? 

An8.  $52.02. 

285.  Rule  for  finding  the  Cost  when  the  Quantity  and  the  Price  of 
a  Ton  of  2000  pounds  are  ^\s^x\.— Multiply  the  jnnce  by  half 
the  number  of  thousands  in  the  quantity. 

PIIOBLJEMS. 

1.  How  much  must  be  paid  for  transporthig  96140  pounds 
of  coal,  at  $1.40  per  ton  ?  Ans.  $67.29|-. 

2.  At  $10  per  ton,  what  will  5154  pounds  of  ground  plaster 
cost? 

3.  A  farmer  bought  50  bags  of  superphosphate  of  lime, 
each  containing  125  pounds,  at  $55  per  ton.  What  did  it 
cost  him  ?  Ans.  $171.87|. 

4.  I  bought  one  load  of  hay  weighing  23456  pounds,  and 
another  weighing  31640  pounds.  What  was  the  cost  of  both, 
at  $31.50  per  ton? 

5.  What  will  be  the  freight  on  27340  pounds  of  coal,  at  $2 
per  ton,  and  on  65400  pounds  of  merchandise,  at  $1.10  per 
ton?  '  Ans.  $63.31. 

6.  How  much  must  be  paid  for  14  loads  of  coal,  each 
weighing  1750  pounds,  at  $6.50  per  ton?  Ans.  $79.62-|. 


128 


UNITED    STATES    MO^EY. 


SECTION    XXXII. 
BILLS   AXD    ACCOUJ^TS. 

286.  A  Bill  of  (joods  is  a  written  statement  of  articles  sold, 
the  quantity  and  price  of  each  article,  and  the  entire  cost  of 
the  whole,  together  with  the  date  of  the  transaction  and  the 
names  of  the  purchaser  and  seller. 

287.  A  Bill  of  Services  is  a  written  statement  of  labor  per- 
formed, and  the  time,  kind  and  value  of  such  services. 

288.  An  Invoice  is  a  full  statement  of  goods  or  merchandise 
forwarded  to  the  purchasers,  with  marks,  numbers  and  con- 
tents of  each  package,  and  the  charges  for  cartage,  insurance,  etc. 

289.  The  Footing  of  a  bill  is  the  total  cost  of  the  items. 

290.  A  Debtor  is  the  party  who  owes  a  debt,  and  a  Creditor 
is  the  party  to  whom  the  debt  is  owed. 

291.  A  bill  is  Receipted  when  the  creditor,  or  some  one  act- 
ing for  him,  acknowledges  its  payment  in  writing. 

292.  An  Account  is  an  entry  or  record  of  items  of  debt  and 
credit  between  parties. 

293.  A  Statement  of  an  Account  is  a  bill  of  the  items  of  an 
account. 

In  bills  and  accounts  the  character  @  signifies  at.  C.  is  often  used  for 
hundreds,  and  M.  for  thousands. 


WRITTEK    EXERCISES. 

294.  Bill  Unreceipted. 

New  York,  Jan.  5,  1871. 
Mr.  George  W.  Lane, 

Bought  of  James  Ross  &  Co. 


20  yd.  Calico, 

@ 

$  .17' 

$3 

40 

12    "    Gingham, 

@ 

.40 

4 

80 

14-    "    Cambric, 

@ 

.25 

3 

50 

Find  the  footing  of  this  bill. 


UNITED  STATES  MONET. 


129 


Bill  lieceipted. 

Philadelphia,  March  4,  1871. 
Dr.  Charles  N.  Thayer, 

Bought  of  T.  B.  Grimshaw  &  Co. 


15  lb.  Japan  Tea,       @ 

$1.20 

$ 

10    ''  Bio  Coffee,        @ 

.35 

20  gal.  Mola^sses,         @ 

.80 

2  hush.  Corn  Meal,  @ 

1.00 

Received  payment, 

T.  B.  Grimshaw  &  Co. 
What  is  the  footing  of  this  bill  ? 


Sill  Receipted  hij  Clerk, 

Chicago,  Dec.  27,  1871. 
Messrs.  Reed  &  Fuller, 

To  William  Ogden,  Dr. 


1870. 

Oct. 

4 

To  6  pr.  Calf  Boots,            @^6.00 

# 

11 

w 

"    8    "  Kip  Boots,             @    S.oO 

Nov. 

6 

"  20  "  Ladies'  Kid  Shoes,  %     1-60 

a 

« 

"    3  doz.  Men's  Hats,          @  18.00 

Dec. 

29     "    6  pr.  Misses' Kid  Boots,  @     2.25 

# 

Received  payment, 

William  Ogden, 

per  L.  T.  Smith. 
What  is  the  footing  of  this  bill  ? 


130  UNITED  STATES   MONET. 

Bill  of  Services. 

Boston,  Jan.  6,  1871. 
Maj.  John  Stone, 

To  Henry  L.  Doten,  Dr. 


1870. 

Nov. 

11 

For  Labor  of  myself  and  apprentice, 

on  House, 

$7 

50 

Dec. 

3 

"       "    onBarn,  10  days,@$3.50 

85 

$4^ 

00 
50 

Received  payment  by  Note, 

Henry  L.  Doten. 
What  is  the  footing  of  this  bill  ? 


Statement  of  Account. 

New  Orleans,  Nov.  1,  1871. 
Col.  LoiTis  Lamert, 

To  John  Degan  &  Co.,  Dr. 


1871. 
Aug. 

12 

<( 

16 

Sept. 

13 

Aug. 

15 

Oct. 

9 

To  15650  ft.  Pine  Boards,  No.  1, 
%  $80.00  p)er  M., 

"  4^00  ft.  Clapboards,  extra, 
@  $60.00  per  M., 

"  6350  ft.  Pine  Lumber,  No.  3, 
@  $65.00  per  3L, 

Or. 
By  Merchandise,  as  by  his 

bill,  $750.00 

"     Cash,  950.00 

Balance  due  J.  D.  &  Co., 

Received  payment, 

John  Decjan  &.  Co. 

What  is  the  balance  of  the  account  diu^  J.  D.  &  Co.  ? 


s 


1700 


UNITED    STATES    MONET.  131 

6.  James  Riley  &  Co.  bought,  July  7,  1870,  of  Joseph  Herr, 
Trentou,  N.  J.,  15  toua  of  coal,  at  $6.50  per  tou;  19  tons  of 
coal,  at  $5.25  per  ton ;  and  14|-  cords  of  wood,  at  $7.25  per 
cord.  Make  a  bill  of  the  purchase,  find  the  footing,  and  re- 
ceipt the  bill  for  Joseph  Herr.  Ans.  Footing,  $302.37^. 

7.  Feb.  17,  1870,  Patrick  Mahoney  bought  of  James  Bur- 
rows, Harrisburg,  Pa.,  3  barrels  of  flour,  at  $9.50  a  barrel ; 
20  pounds  of  cotlee,  at  30  cents  a  pound ;  and  25  pounds  of 
sugar,  at  17-|-  cents  a  pound.  Feb.  18,  1870,  he  paid  $20  ;  and 
March  3,  1870,  he  rendered  a  bill  for  3  days'  work,  at  $1.75  a 
day.  Put  this  in  the  form  of  an  account,  and  find  the  balance 
due  James  Burrows.  Ans.  $13.62^. 

TEST    QUESTIONS. 

29o. — 1.  What  is  Money?  How  does  paper  money  differ  from  coin 
or  specie  ?  What  is  the  money  in  common  circuhition  called  ?  What  is 
United  States  money  ?     Recite  the  table  of  United  States  money. 

2.  Of  what  are  the  Coins  of  the  United  States  made  ?  What  coins 
are  of  gold?     Of  silver?     Of  nickel?     Of  bronze? 

.3.  What  is  the  money  of  the  Dominion  of  Canada  called  ?  Of  what 
does  it  consist  ?     What  ar©  the  Canada  corns  ? 

4.  What  is  the  Unit  of  United  States  money  ?  How  may  dimes, 
cents  and  mills  be  decimally  expressed?  Why  is  United  States  money  a 
decimal  currency  ?  How  are  dimes  commonly  regarded  ?  How  may  a 
decimal  of  a  dollar  less  than  a  cent  be  read  ? 

5.  What  is  Denomination?  Of  two  denominatiions  which  is  the 
higher?  What  is  reduction ?  How  are  cents  reduced  to  mills ?  Dollars 
to  mills?  Dollars  and  cents  to  cents?  Dollars,  cents  and  mills  to  mills? 
Mills  to  cents  ?     Mills  to  dollars  ? 

6.  How  are  Computations  in  United  States  money  performed  ?  What 
is  quantity  in  the  sale  of  property  ?  What  is  price  ?  What  is  cost  ? 
What  are  aliquot  parts  of  a  number  ?  What  aliquot  parts  of  a  dollar 
are  frequently  used  in  business? 

7.  How  is  the  Cost  found  when  the  price  at  an  aliquot  part  of  a  dol- 
lar and  the  quantity  are  given  ?  When  the  price  of  100  or  1000,  and  the 
quantity  are  given  ?  When  the  price  of  a  ton  of  2000  pounds,  and  the 
quantity  are  given? 

8.  What  is  a  Bilt.  of  Goods?  A  bill  of  services?  An  invoice? 
The  footing  of  a  bill  ?  When  is  a  bill  receipted  ?  What  is  an  account  ? 
A  statement  of  an  account? 


132  REVIEW. 


SECTION    XXXIII. 
REVIEW  PROBLEMS. 

WRITTEN  EXERCISES. 

296. — Ex.  1.  I  gave  .15  of  a  sum  of  money  in  charity,  .375 
of  it  for  payment  of  debts,  and  .125  of  it  for  a  library.  What 
fractional  part  of  the  sum  had  I  left  ?  Ans.  ^. 

2.  What  common  fraction  is  equivalent  to  .41f  ?     Ans.  -^2. 

3.  What  will  be  the  cost  of  610  yards  of  cloth,  at  $1.37| 
per  yard  ? 

/)<$rin  nn  Solution.— $1.37^  is  equal  to  $1.+  $.25  +  $.12^. 

4J^ blU.UU  610  yards  at  %l  per  yard  will  cost  $610 ;  at  $.25,  or  $^, 

2)152.50  will  cost  }  of  $610,  or  $152.50;  and  at  $.124,  or  J  of 

76.25  $i  will  cost  \  of  $152.50,  or  $76.25.     Hence,  610 

rftooj?  /yr  yards  at  $1,374  per  yard  will  cost  $610  +  $152.50  + 

^ii^^.Ib  $76.25,  or  $838:75. 

4.  How  much  must  be  paid  for  9870  bushels  of  wheat,  at 
$.871  per  bushel  ?  Ans.  $8636.25. 

5.  If  you  should  pay  $73.50  for  a  wagon,  and  3  times  as 
much  for  a  horse,  what  would  be  the  cost  of  both  ? 

6.  The  annual  fall  of  rain  in  Boston  is  39.23  inches ;  in 
Providence,  36.74  inches;  in  New  York,  36  inches;  and  in 
Washington,  34.62  inches.     What  is  the  average  ? 

7.  How  much  can  you  save  in  a  month  of  26  working  days, 
if  you  earn  $1.75  and  spend  62^  cents  each  day? 

8.  What  is  the  cost  of  4565  feet  of  joist  at  $23  per  M.,  and 
13640  feet  of  boards  at  3535.50  per  M.  ? 

9.  The  annual  fall  of  rain  in  Savannah  is  55  inches,  and  for 
each  inch  the  volume  of  rain  is  17.3387  million  gallons  per 
square  mile.  What  is  the  volume  of  rain  in  Savannah  for  a 
square  mile?  Ans.  953.6285  million  gallons. 

10.  A  merchant  bought  molasses  at  855.45  per  cask,  and 
sold  it  at  $63.70  per  cask,  thereby  gaining  $858.  How  many 
casks  did  he  buy  ?  Ans.  104. 

11.  A  fire  destroyed  .375  of  a  quantity  of  goods  worth 
$2000.  What  sum  did  a  man  lose  who  owned  .25  of  the 
whole  ? 


REVIEW.  133 

12.  What  is  the  net  profit  of  a  garden,  according  to  the 
following  items? — 

Expenses. — Paid  for  plowing  and  carting,  $31 ;  for  fertilizers 
and  seeds,  $77.15  ;  for  labor  and  cultivating,  $31 ;  and  for  cost 
of  harvesting  and  marketing,  $43.18. 

Income. — Sold  56  bushels  of  potatoes,  at  $1.50  ;  13  bushels 
of  peas,  at  $3 ;  70  dozen  cabbages,  at  $1.12|^ ;  50  dozen  green 
corn,  at  $.37^;  and  170  boxes  of  strawberries,  at  30  cents. 

Ans.  Net  profit,  $89.17. 

13.  Henry  Mclntire  of  Nashville  has  worked  for  Jacob 
Sangster  5  days,  at  $4.25  per  day ;  and  he  has  furnished  5000 
bricks,  at  $13.25  per  thousand,  and  3  casks  of  lime,  at  $1.80 
per  cask.  Make  out  the  statement  of  the  account,  and  find 
the  amount  due. 

SECTION    XXXIV. 
DEJ^OMIJfA  TE  JV UMBERS. 

297.  Quantity,  in  general,  is  that  which  admits  of  measure- 
ment or  computation. 

Thus,  distance,  space,  etc.,  which  can  be  measured,  and  numbers, 
money,  etc.,  which  can  be  computed,  are  quantities.  "^ 

298.  A  Measure  is  a  unit  used  in  estimating  or  determining 
quantity. 

299.  A  quantity  is  measured  by  finding  how  many  times  the 
quantity  contains  the  unit. 

Thus,  the  length  of  a  table  is  measured  or  determined  by  applying  a 
foot-rule,  and  thus  finding  how  many  times  the  length  of  the  rule  is  con- 
tained in  the  length  of  the  table. 

MEASURES  OF   EXTENSION. 

300.  Extension  is  that  which  has  one  or  more  of  the  dimen- 
sions, length,  breadth  and  depth,  or  thickness. 

301.  A  Line  is  that  which  has  only  length. 

302.  A  Surface  is  that  which  has  only  length  and  breadth. 

303.  A  Solid,  or  Volume,  is  that  which  has  length,  breadth 
and  thickness,  or  depth. 

12 


134 


DENOMINA  TE  N  UMBERS. 


LINEAR    MEASURES. 

304.  Linear  or  Long  Measures  are  those  used  in  ascertain  iug 
lengths  and  distances. 

TABLE. 

12  inches  (in.)       are  1  foot ft. 

3  feet  "    1  yard.  ...yd. 

5 J  yards,  or  16^  ft."     1  rod rd. 

320   rods  "    1  mile.  . . .  mi. 

1  mi.  =  320  rd.  =  17 GO  yd.  =  5280  ft. 

305.  In  Cloth  Measure  the  linear  yard  is  divided  into  Imhes, 
quarters,  eighths  and  sixteenths,  or  iwik. 

2  si.vteenths,  or  2  nails,  are  1  eighth  .  .  .  8th. 
2  eighths,  or  ^  na,ils          "    1  quaTter  .  .  qr. 
4  quarters  "     1  yard, yd. 

306.  The  Surveyor's  Chain,  called  Ghmter's  chain,  used  in 
measuring  roads  and  boundaries  of  land,  is  4  rods  in  length, 
and  is  subdivided  as  follows : 

7.92  inches  are  1  link.  .  .  .  li. 

100  links,  or  Jf  rods,  "    1  chain  .  .  ch. 

80  chains  "    1  niile  .  .  .  7;?/, 

Engineer!^  Measuring  Tape,  usi'd  in  ineasnriiiEj  railroads  aiul  canals  is 
100  feet  in  length,  witli  eacli  foot  divided  into  tenths. 


DE NOMINA  TE  NUMBERS. 


135 


In  measuring  ropes,  cables  and  short  sea-distances,  6  feet  are  1  Fathom, 
and  120  fathoms  are  1  Cable- Length.  In  measuring  longer  sea-distances, 
1.15  +  miles,  or  ^^^  of  the  average  length  of  degrees  measured  on  a 
meridian  of  the  earth,  is  a  Geographic  or  Nautical  Mile,  or  Knot,  and  3 
nautical  miles,  or  3.45  +  common  miles,  are  1  Nautical  League. 

A  Furlong  is  40  rods,  but  this  term  is  becoming  obsolete. 

A  Meier  is  39.37  inches;  a  Kilometer,  or  1000  meters,  is  .62137  of  a 
mile ;  a  Centimeter,  or  j-^g  of  a  meter,  is  .3937  of  an  inch. 

307.— Ex.  1.  How  many  inches  are  10  feet?     Are  12  feet? 

2.  In  48  inches  how  many  leet  ?   In  96  inches  how  many  feet  ? 

3.  How  many  feet  in  2  rods  ?     In  3  rods  ? 

4.  What  part  of  a  mile  is  40  rods  ?     Is  80  rods  ? 

5.  How  many  rods  are  33  feet  ?     Are  22  yards  ? 

SURFACE    MEASURES. 

308.  A  Straigrht  Line  is  a  line  that  has  only  one  direction. 
Thus,  the  line  AB  is  a  straight  line.  ■'^ '^ 

309.  An  Angle  is  the  difference  of 
direction  of  two  lines  drawn  from  the 
same  point. 

Thus,  the  lines  AB  and  AC,  meeting  at  A, 
form  the  angle  CA  B. 

310.  A  Perpendicular  Line  is  a  straight  line  meeting  another 
straight  line  so  as  to  form  two  equal  angles.  d 

Thus,  the  line  CD  is  a  perpendicular  line. 

311.  A  Right  Angle  is  an  angle  formed 
by  two  lines  perpendicular  to  each  other. 

Thus,  the  angles  A  CD  and  DCB  are  each  right 
angles. 


312.  A  Square  is  a  figure  having  four 
equal  straight  sides  and  four  equal  angles. 

Thus,  a  Square  Inch  is  a  square  having  each 
of  its  sides  1  inch  in  length. 


A  Sijuare  Inch. 


136  DENOMINATE  NUMBERS. 

313.  A  Rectangle  is  any  figure  having  four  straight  sides 
and  four  equal  angles. 

Thus,    the  figure  in   the  margin,  having  four  straight 
sides  and  four  equal  angles,  is  a  rectangle. 


314.  Surface  or  Square  Measures  are  those  used  in  ascertain- 
ing the  extent  of  surfaces. 


TABLE. 

IJfJj.  square  inches  (sq.  in.)  en 

9  square  feet 
30-j^  square  yards 
160  square  rods 
6JfO  acres 


e  1  square  foot .  .  sq.  ft. 
1  square  yard  .  sq.  yd. 
1  square  rod  .  .  sq.  rd. 

1  Acre A. 

1  square  mile  .  sq.  mi. 


1  A.  =  160  sq.  rd.=4840  sq.  yd.  =43560  sq.  ft.= 
6272640  sq.  in. 

315.  In  the  Measurement  of  Laud  the  square  chain  and  its 
subdivisions  are  used. 

625  square  linhs  (sq.  li.)  are  1  square  i^od sq.  rd. 

16  square  rods                    "    1  square  chain  .  .  .  sq.  eh. 
10  square  chains  "    1  Acre A. 

A  Perch,  of  surface  is  a  square  rod,  and  a  Rood  is  40  square  rods. 
These  denominations  are  now  seldom  used. 

A  Section  of  land  is  a  square  mile,  and  a  Quarter  Section  is  160  acres. 

A  Square  in  architects'  measure  is  100  square  feet. 

A  Square  Meter,  or  a  Centiare,  is  1550  square  inches,  or  1.196  square 
yards  ;  an  Are  is  119.6  square  yards ;  and  a  Hectare,  or  100  ares,  is  2.471 
acres. 

316. — Ex.  1.  How  many  square  feet  are  9  square  yards? 

2.  In  81  square  feet  how  many  square  yards? 

3.  How  many  square  rods  in  10  square  chains  ? 

4.  In  160  square  rods,  how  many  square  chains  ? 

5.  What  part  of  an  acre  is  80  square  rods?  Is  120  square 
rods? 

6.  How  many  square  rods  in  one  fuurtli  of  an  acre? 


DENOMINATE  NUMBERS. 


137 


CUBIC    MEASURES. 

317.  A  Cube  is  a  solid 
bounded  by  six  equal 
squares,  called  its  faces. 

318.  A  Cubic  Inch  is  a 

cube  whose  faces  are  each 
1  square  inch. 

319.  A  Rectangular  Solid 

is  any  volume  bounded  by 
six  rectangular  faces. 


A  Cubic  Inch. 


A  Rectangular  Solid. 


Thus,  the  figure  in  the  margin,  having 
six  rectangular  faces,  is  a  rectangular 
solid. 

320.  Cubic  Measures  are  those 
used  in  measuring  things  that  have 
length,  breadth  and  depth,  or  thick- 
ness. 

TABLE. 

1728  cubic  inches  (cii.  in.)  are  1  cubic  foot. . .  cu.  ft. 
27  cubic  feet  "    1  cubic  yard. .  cu.  yd. 

1  cu.  yd.  =  27  cu.  ft.  =  ^6656  cu.  in. 

321.  In  Measuring'  Wood  the  cord  foot  and  cord  are  used. 

Thus, 

16  cubic  feet  are  1  coj^d  foot cd.  ft. 


8  cord  feet,  or  \ 


•e  1  cord, cd . 


128  cubic  feet, 

A  Ton,  in  computing  the  tonnage  of  ships  and  other  vessels,  is  100  culiic 
foet  of  their  internal  space. 

A  Ton  of  freight,  for  some  articles,  is  estimated  by  the  space  occupied. 
The  heavier  articles  are  estimated  by  their  weight. 

A  Perch  of  stone  is  16|  feet  long,  1  foot  high  and  1^  feet  thick,  and 
contains  24|  cubic  feet. 

A  Cubic  Meter,  or  Stere,  is  35.316  cubic  feet,  or  .2759  of  a  cord. 
12  * 


138 


DEXOMIXA  TE  NUMBERS. 


Wood,  as  usually  cut  for  market,  is  4  feet  long,  and  is  piled  in  ranges 
4  feet  high.  Of  such  ranges,  a  part  that  is  1  foot  of  the  length  of  the 
range  is  1  cord  foot ;  a  part  that  is  8  feet  of  the  length  is  1  cord. 


322.— Ex 

4^6  rd. 
5i 


WUITTEN   EXERCISES. 

1.  How  many  feet  are  46  rods  ? 


Solution. — Since  1  rod  is  5|  yards,  there  must  be  5| 
times  as  many  yards  as  rods ;  hence,  in  46  rods  there  must 
be  5^  times  46  yards,  or  253  yards. 

Since  1  yard  is  3  feet,  there  must  be  3  times  as  many  feet 
as  yards ;  hence,  in  253  yards  there  are  3  times  253  feet,  or 
759  feet,  which  is  the  answer  required. 


How  many  rods  are  759  feet  ? 

Solution. — Since  3  feet  are  one  yard, 
759  feet  must  be  as  many  yards  as  there 
are  times  3  feet  in  759  feet,  which  are  253 
times.     Hence,  759  feet   =  253  yards. 

Since  5j  yards  are  1  rod,  253  yards  must 

be  as  many  rods  as  there  are  times  51 

yards   in    253    yards,    or    times   11    half 

yards   in   506  half  yards,  which   are  46 

Hence,  253  yards,  or  759  feet,  =r.  46  rods. 


S  ft.  )  759  ft. 

5 1  yd.  )  253  yd. 
2_  _^ 

llhf.yd.)506  hj.yd 
4-6  rd. 


times, 

3.  How  many  rods  are  45  mile.s? 


DENOMINATE  NUMBERS.  139 

4.  How  many  miles  are  14400  rods? 

5.  How  many  square  feet  are  110  acres? 

6.  How  many  acres  are  4791600  square  feet? 

7.  How  many  square  links  are  5  acres  ? 

8.  How  many  acres  are  500000  square  links  ? 

9.  How  many  cubic  feet  in  312  cubic  yards? 

10.  What  will  25  cords  of  wood  cogt,  at  75  cents  per  cokI 
foot  ? 

MEASURES  OF  CAPACITY. 

323.  Measures  of  Capacity  are  measures  used  in  ascertaining 
the  quantity  of  liquids,  and  some  dry  articles. 

LIQUID    MEASURES. 

324.  Liquid  Measures  are  those  used  in  measuring  liquids. 

TABLE. 

^  gills  (gi.)  are  1  pint pt. 

2  pints  "    1  quart.  .  .  .  qt, 

^  quarts         "    1  gallon  .  .  ,  gal. 

Igal.  =4qt.  =  8  pt.  =  32  gi. 

The  United  States  Standard  Gallon  contains  231  cubic  inches;  the  Ln- 
perial  Gallon  of  Great  Britain  contains  277.274  cubic  inches. 

Malt  liquors  and  milk  were  formerly  sold  by  what  was  called  Beer 
Measure,  the  gallon  of  which  contained  282  cubic  inches. 

A  Barrel,  regarded  as  a  measure  of  cisterns,  vats,  etc.,  is  31^  gallons, 
and  a  Hogshead  is  63  gallons ;  but  these  terms  in  commerce  are  often 
applied  to  casks  of  various  capacities. 

325.  Apotliecaries'  Fluid  Measures,  or  those  used  in  com- 
pounding medicines  and  in  putting  up  medical  prescriptions, 
are  as  follows — 

60  minims  (tn)  are  1  fluid  dram fz. 

8  fluid  drains       "    1  fluid  ounce /o. 

16  fluid  ounces      "    1  pint 0. 

8  pints  "    1  gallon Cong. 


140  DENOMINATE  NUMBERS. 

326. — Ex.  1.  How  mauy  gills  in  3  quarts  ? 

2.  In  40  gills,  how  mauy  quarts  ? 

3.  How  many  pints  in  (5  gallons  ? 

4.  In  40  pfets,  how  many  gallons  ? 

5.  What  will  4  gallons  of  milk  cost,  at  o  cents  a  quart  ? 

6.  How  many  gallons  of  milk,  at  5  cents  a  quart,  can  be 
bought  for  60  cents  ?  , 

DRY   MEASURES. 

327.  Dry  Measures  are  those  used  in  measuring  dry  articles, 
such  as  grain,  fruit,  vegetables,  coals,  etc. 

TABLE. 

2 pints  (pt.)  are  1  r/uart.  .  .  qt. 

-  8  quarts  "    1  peck.  .  .  .  ph. 

^  pedes  "    1  bushel .  .  bu. 

lbu.=4  pk.  =  S2  qt.  =  64  pt. 

The  United  Stales  Standard  Bufshel  contains  21o0.42  cubic  inches.  The 
Imperial  Bushel  of  Great  Britain  contains  2218.192  cubic  inches. 

Four  Heaped  Pecks  are  equal  to  5  even  pecks  ;  and  6  quarts  of  dry 
measure  are  very  nearly  equal  to  7  liquid  quarts. 

A  Liter  is  1.0567  liquid  quarts,  or  .908  of  a  dry  quart,  or  61.022  cubic 
inches ;  and  a  He<:toliter,  or  100  liters,  is  2  bushels  3.35  pecks,  or  3.531 
cubic  feet.    A  Centiliter,  or  j^^  of  a  liter,  is  .338  of  a  fluid  ounce. 

328. — Ex.  1.  How  many  quarts  are  6  pecks  ? 

2.  In  64  quarts,  how  many  bushels  ? 

3.  At  15  cents  a  peck,  what  wall  1  bushel  of  apples  cost? 

4.  How  many  pecks  of  apples,  at  60  cents  a  bushel,  can  be 
bought  for  45  cents  ? 

WJtITTEN   EXEKCISES. 

329. — Ex.  1.  How  many  quarts  in  6  hogsheads? 

2.  In  1512  quarts,  how  many  hogsheads? 

3.  How  many  quarts  in  112  bushels? 

4.  In  3584  (|uarts,  how  many  l)ushcls  ? 

5.  What  will  5  busliels  of  chestnuts  cost,  at  6  cents  a  pint  ? 


LENOMISA  TE  LUMBERS. 


141 


MEASURES  OF  WEIGHT. 

330.  Weight  is  the  quantity  of  matter  iu  bodies,  as  deter- 
mined by  the  force  with  which  they  tend  toward  the  earth. 


I  I  m 

I         --. 
'      CORH 


AVOIRDUPOIS    WEIGHTS. 


331.  Avoirdnpois  Weiglits  are  those  used  in  weighing  pro- 
duce, groceries,  iron,  etc. 

TABLE. 

16  ounces  (oz.)  are  1  pound lb. 

100  pounds  "    1  hundred-iv  eight ..  .Gwt. 

20  hundred-iu eight     "    1  ton T. 

IT.  =  20  cwt.  =  2000  lb.  =  32000  oz. 

332.  In  collecting  duties  upon  foreign  goods,  at  the  United 
States  Custom  Houses,  and  also  in  freighting  coal,  and  selling 
it  by  wholesale, 

28  pounds  are  1  quarter. 

Jf,  quarters,  or  112  pounds ,  "    1  hundred-weight. 

20  hundred-weight. or  2240 lbs.,"  1  long  ton. 

The  ounce  is  considered  as  16  drams,  but  the  dram  is  not  recognized 
in  business. 

The  term  Cental  is  beginning  to  be  used  for  the  hundred-weight. 
A  Quarter  of  Gh-ain  in  Great  Britain  is  560  pounds,  or  8  Imperial  bushels. 
A  Oram  is  .03527  of  an  ounce;  a  Kilogram,  or  1000  grams,  is  2.2046 
pounds;  and  a  Tonneau,  or  100  kilos,  is  2204.6  pounds. 


142 


DEA^OMINATE  NUMBERS. 


333.  The  following  Units  are  sanctioned  by  custom  or  law — 


32  lb.  of  oats 

Jf5  lb.  of  timothy -seed 

^8  lb.  of  barley 

56  lb.  of  rye 

56  lb.  of  Iiidiaib  com 

50  lb.  of  Indian  meal 

60  lb.  of  wheat 

60  lb.  of  clover -seed 

60  lb.  of  potatoes 

56  lb.  of  butter 
100  lb.  of  meal  or  flour 
100  lb.  of  grain  or  flour 
100  lb.  of  dry  fish 
100  lb.  of  nails 
196  lb.  of  flour 
200  lb.  of  beef  or  pork 


are 


bushel. 

bushel. 

bushel. 

bushel. 

bushel. 

bushel. 

bushel. 

bushel. 

bushel. 

fij^hin. 

sack. 

cental. 

quintal. 

cask. 

barrel. 

barrel. 


334. — Ex.  1.  How  many  ounces  in  5  pounds? 

2.  In  64  ounces,  how  many  pounds  ? 

3.  How  many  pounds  in  -j%  of  a  hundred-weight? 

4.  How  many  pounds  in  f  of  a  ton  ? 

5.  What  part  of  a  hundred-weight  is  70  pounds  ? 

6.  How  much  must  be  paid  for  f  of  a  pound  of  spice,  at  3 
cents  an  ounce? 

7.  How  much  nuist  be  paid  for  |-  of  a  quintal  of  fish,  at  8 
cents  a  pound  ? 

8.  How  many  pounds  in  3  pecks  of  clover-seed  ? 

9.  How  much  will  a  barrel  of  beef  cost,  at  10  cents  a  pound  ? 

10.  How  much  will  a  cental  of  flour  cost,  at  h\  cents  a 
pound  ? 

11.  How  much  will  a  peck  of  timothy-seed  cost,  at  12  cents 
a  pound  ? 

12.  How  much  will  a  ton  of  meal  cost,  at  $1.12^  per  bushel  ? 

13.  What  will  be  the  cost  of  ten  barrels  of  beef,  at  ten 
cents  per  pound  V 


DENOMINATE   NUMBERS.  143 

TROY  WEIGHT. 

335.  Troy  Weights  are  those  used  in  weighing  gold,  silver 

and  gems. 

TABLE. 

24-Mrnins  {£r.)     are    1  pennyweight ....  pwt. 

20  pennyweights  "      1  ounce oz. 

12  ounces  "      1  pound lb. 

llb.==12  oz.=240  pivt.=5760  gr. 

336.  Apothecaries,  in  compounding  medicines  and  in  put- 
ting up  medical  prescriptions,  either  use  only  the  units  of 
grains,  ounces  and  pounds,  or  subdivide  the  Troy  pound. 
Thus — 

20  gj^ains  {gr.)    are  1  scruple 9. 

S  scruples  "     1  dram .5, 

8  drains  "     1  ounce oz. 

1%  ounces  "     1  pound Ih. 

An  ounce  avoirdupois  is  437|  grains,  and  an  ounce  Troy  is  480  grains. 
A  pound  avoirdupois  is  7000  grains,  and  a  pound  Troy  is  5760  grains. 
A  Oram  is  15.432  grains  Troy. 

337. — Ex.  1.  How  many  ounces  in  5  pounds  Troy? 

2.  In  72  ounces  Troy,  how  many  pounds? 

3.  How  many  pennyweights  are  8  ounces  ? 

4.  How  many  ounces  are  200  pennyweights  ? 

5.  How  much  will  |  of  a  pennyweight  of  metal  cost,  at 
12  cents  a  grain? 

WniTTEIf  EXERCISES. 

338. — Ex.  1.  How  many  ounces  are  35  tons? 

2.  In  1120000  ounces,  how  many  tons? 

3.  How  many  grains  are  15^  pounds  Troy? 

4.  In  89280  grains  Troy,  how  many  pounds  ? 

5.  What  will  2565  pounds  of  coal  cost,  at  $8  per  ton  ? 

6.  What  is  the  value  of  11  pounds  of  fine  silver,  at  $.06f 
per  pennyweight  ?  Ayis.  $179.52. 

7.  How  many  pounds  of  fine  silver  can  be  bought  for 
$179.52,  at  $.06|-  per  pennyweight? 


144 


DEyOMIXA  TE  XUMBERS. 


MEASURES  OF  TIME. 

339.  Time  is  a  definite  portion  of  duration. 

TABLE. 

60  seconds  [sec),  are  1  minute min. 

60  minutes  "      1  hour h. 

2Jj.  hours  "      1  day d. 

36o  days  "      1  common  year .  .  .y. 

S66  days  "      1  leap  year. 

Also, 

7  dajjs  are  1  week wk. 

52  weeks  1  day     "    1  common  year  .  .  .y. 

100  years  "     1  century cen. 

\  sec 
ly.  =  365  d.^8760  h.-=525,600  min.  =3 1,5 36, 000 

340.  The  Months,  their  names,  and  the  number  of  days  in 
each,  are — 

Days. 

1st  mo.,  January,  has  31. 


2d  mo. 

February,  ' 

28  or  29. 

3d  mo. 

March, 

31. 

4th  mo. 

April,         ' 

30. 

5th  mo. 

May, 

31. 

6th  mo. 

June,          ' 

30. 

has 


7th  mo.,  July, 

8th  mo.,  August,       " 

9th  mo.,  September,  " 

10th  mo.,  October,      " 

11th  mo.,  November,  " 

12th  mo.,  December,  " 


Days. 

31. 
31. 
30. 
31. 
30. 
31. 


341.  The  exact  length  of  the  year,  or  the  precise  time  in 
which  the  earth  makes  one  revolution  around  the  sun,  is  365d. 
5h,  48min.  49.7sec.,  or  nearly  365^  days.  Hence,  on  account 
of  this  fractional  part  of  a  day,  at  certain  points  of  time  Feb- 
ruary has  29  days,  and  the  year  is  leap  year. 

Every  centennial  year  mhose  number  is  exactly  divisible  by  400, 
and  every  year  not  a  centennial  year  whose  number  is  exactly 
divisible  by  4,  w  leap  year,  or  has  o()(j  days. 

342. — Ex.   1.  How  many  hours  in  f  of  a  day? 

2.  What  part  of  a  minute  is  55  seconds  ? 

3.  How  many  months  have  30  days  each  ? 


DENOMINATE   NUMBEliS.  145 

4.  lu  65  days,  how* many  weeks,  and  what  number  of  days 
remain  ? 

5.  From  April  1  to  September  1,  how  many  months? 

6.  How  much  can  be  earned,  at  12  cents  per  hour,  in  6 
working  days  of  10  hours  each  ? 

7.  From  January  12  to  Marcli  12,  in  a  common  year,  how 
many  days  ? 

WlilTTEN    EXERCISES. 

343. — Ex.  1.  How  many  seconds  in  48  hours? 

2.  How  many  hours  in  172800  seconds  ? 

3.  How  many  hours  in  16  years,  allowing  every  fourth  year 
to  be  a  leap  year  ? 

4.  How  many  years,  of  3651  days  each,  are  in  140256  hours  ? 

5.  How  many  months  in  5  centuries? 

6.  How  many  centuries  in  6000  months  ? 

7.  How  much  will  a  man  earn  in  3  years,  at  $56  per  month  ? 


MEASURES  OF  CIRCLES. 

344.  A  Circle  is  a  plane  surface,  which  is  bounded  by  a  line 
having  all  its  parts  equally  distant  from 
a  i^oint  within,  called  the  center. 

345.  A  Circumfereuce  is  the  line  that 
bounds  a  circle. 

346.  An  Arc  is  any  part  of  the  cir- 
cumference of  a  circle;  as  AD,  DB  or 
BA. 

347.  A  Diameter  of  a  circle  is  a  straight  line  drawn  through 
the  center  of  that  circle,  and  terminated  both  ways  by  the  cir- 
cumference ;  as  the  straight  line  AB. 

348.  A  Degree  is  one  of  the  360  equal  parts  of  a  circum- 
ference. 

349.  The  Measure  of  an  Angle,  whose  sides  meet  at  the 
center  of  a  circle,  is  that  part  of  the  circumference  included 
between  the  sides. 

Thus, the  arc  AD  is  the  measure  of  the  angle  ACD. 
13 


146 


DENOMINA  TE  NUMBERS, 


350.  Circular  and  Ang-ii- 
lar  Measures  are  those  used 
for  measuring  angles  and 
the  difference  of  directions 
and  in  determining  latitude 
and  longitude,  etc. 

Seconds  are  usually  subdivided 
into  tenths  or  hundredths. 

A  SexUivt  is  one  sixth  of  a 
circumference,  a  Quadrant  one 
fourth,  and  a  Semi-circumference 
one  half.  A  Minute  of  the  cir- 
cumference of  the  earth  is  a 
geographic  mile. 


TABLE. 

60  seconds  ("  )  are  1  minute '. 

60  minutes  "     1  degree °. 

S60  degrees  "    1  circumference C. 

1  a-^ 360°  -^21600' =  1296000''. 

351.  In  Astronomy,  the  zodiac,  or  the  apparent  path  of  the 
sun  in  the  heavens,  is  divided  into  12  equal  parts,  called  signs 
Hence, 

30  degrees    are    1  sign S. 

12  signs  "      1  circumference C. 

Degrees  of  the  earth's  circumference  on  a  meridian  average 
69.16  common  miles. 

Every  circle,  great  or  small,  has  the  same  number  of  de- 
grees, minutes  and  seconds  ;  hence,  these  parts  of  different 
circles  have  different  lengths. 

352. — Ex.  1.  How  many  minutes  are  3  degrees? 

2.  In  240  degrees,  how  many  minutes? 

3.  What  part  of  a  minute  is  45' seconds? 

4.  How  many  degrees  are  f  of  a  circumference? 

5.  What  part  of  a  degree  is  50  minutes  ?  What  part  of  a 
circumference  is  1 80  degrees  ? 


DENOMINA  TE  NUMBERS. 


147 


MISCELLANEOUS    MEASURES. 


353.  In  the  Papei*  busi- 
ness, the  units  used  are 
given  in  the  following 

TABLE. 

^Jf.  sheets  are  1  quire. 
"BO quires  "   Iream. 
2  reams      "   Ihundle. 
5 bundles  "    Ibale. 

35-4.  In  CoHuting'  cer- 
tain articles,  use  is  made 
of  the  followiuor 


TABLE. 

12  ones    are    1  dozen doz. 

12  dozen  "      1  gross gro. 

12  gross    "      1  great  gross.  ...  grt.  gro. 

355. — Ex.  1.  How  many  ones  are  1  gross? 

2.  How  many  dozens  is  f  of  a  great  gross  ? 

3.  How  many  quires  are  5  reams  ? 

4.  In  5  bales  of  paper  how  many  reams  ? 

5.  At  %\^  per  dozen,  what  will  2  gross  of  writing-books  cost  ? 

6.  How  many  reams  of  paper,  at  10  cents  a  quire,  can  be 
bought  for  $6  ? 

WlilTTEN  EXERCISES. 

356. — Ex.  1.  In  3  common  years,  how  many  minutes? 

2.  In  15776800  minutes,  how  many  common  years  ? 

3.  By  giving  40  minutes  each  day  for  300  days  to  reading, 
how  many  working  days  of  10  hours  each  will  be  thus  devoted  ? 

4.  How  many  dozen  eggs  can  be  put  in  a  box  which  will 
contain  2880  eggs,  and  what  is  their  value  at  20  cents  a  dozen  ? 

5.  Name  the  leap  years  between  1870  and  1881. 

6.  What  will  15  great  gross  of  buttons  cost,  at  3  cents  a 
button?  Am.  $777.60. 


148  DENOMINATE  NUMBERS. 

TEST    QUESTIONS. 

357. — 1.  What  is  Money?  What  is  coin  or  specie?  Currency? 
Recite  the  table  of  United  States  Money.  How  are  computations  in 
United  States  money  performed  ? 

2.  Wh'at  is  Price ?  Cost?  An  aliquot  part?  Give  the  rule  for  find- 
ing the  cost  when  the  price  is  an  aliquot  part  of  a  dollar.  When  the 
price  is  of  100  or  1000.     When  of  2000  pounds. 

3.  What  is  a  Bill  of  goods?  A  bill  of  services?  An  invoice?  The 
footing  of  a  bill  ?     An  account  ?     A  statement  of  an  account  ? 

4.  What  is  a  Measure?  What  is  quantity?  How  is  quantity 
measured  ? 

5.  What  is  Extension  ?    Aline?     A  surface?     A  volume? 

6.  What  are  Linear  Measures?  Eecite  the  table.  How  is  the 
yard  divided  in  measuring  cloth  ?     What  is  used  in  measuring  roads  ? 

7.  What  are  Surface  or  Square  Measures?  What  is  a  straight 
line  ?  An  angle  ?  A  perpendicular  ?  A  right  angle  ?  A  square  ?  A 
rectangle?  Eecite  the  table  of  square  measure?.  AVhat  is  used  in  the 
measurement  of  land  ? 

8.  What  are  Cubic  Measures?  What  is  a  cube?  A  rectangle? 
Give  the  talkie  of  cubic  measures.     What  are  used  in  measuring  wood  ? 

9.  What  are  Measures  OF  Capacity?  What  are  liquid  measures? 
Give  the  table  of  liquid  measures.  What  are  apothecaries'  fluid  meas- 
ures?    What  are  dry  measures?     Give  the  table  of  dry  measures. 

10. -What  is  Weight?  What  are  avoirdupois  weights?  Recite  the 
table  of  avoirdupois  weights.  What  are  Troy  weights?  Recite  tlie 
table  of  Troy  weights.   How  do  apothecaries  subdivide  the  Troy  pound  ? 

11.  What  are  the  measures  of  Time?  Recite  the  table  of  time. 
What  are  the  months  ?     What  years  are  leap  years  ? 

12.  What  are  Circular  Measures?  What  is  a  circle?  A  circumfer- 
ence? An  arc?  A  diameter?  A  degree?  The  measure  of  an  angle? 
Give  tlie  table  of  circular  measures.      How  is  the  zodiac  divided? 

13;  What  units  arc  used  in  Counting  articles?  Recite  the  table. 
What  units  are  used  in  the  paper  business?     Recite  the  table. 

14.  What  is  the  Divisor  in  division?  (Art.  80.)  What  is  an  exact 
divisor  of  a  number?  (Art.  101.)  The  greatest  common  divisor  of  two 
or  more  numbers?  (Art.  112.)     Of  what  is  it  the  product?  (Art.  113—1.) 

15.  What  is  a  Multiple  of  a  number?  (Art.  119.)  How  do  a  mul- 
tiple and  a  divisor  differ?  Why  may  a  multiple  of  a  number  be  called 
tl>e  dividend  of  that  number?.  What  must  the  least  common  multiple 
of  two  or  more  numbers  contain?  (Art.  122.) 


COMPOUND   NUMBERS.  149 

SECTION    XXXV. 

BEDUCTIOJi  OF  COMPOUjYD  A'UMBUES. 

358. — Ex.  1.  How  many  inches  in  2  feet  6  inches?  In"  3 
feet  7  inches  ?     In  2  yards  1  foot  ? 

2.  How  many  feet  in  30  inches  ?  In  43  inches  ?  How  many 
yards  in  84  inches  ? 

3.  How  many  gills  in  2  quarts  1  pint  ?  In  1  quart  1  pint 
1  gill?     In  3  quarts  1  pint? 

4.  How  many  quarts  in  20  gills?    In  13  gills?    In  28  gills? 

5.  How  many  ounces  in  5  pounds  6  ounces?  In  3  pounds 
8  ounces?     In  10  pounds  10  ounces? 

6.  How  many  pounds  in  86  ounces?  In  56  ounces?  In 
170  ounces? 

DEFINITIONS. 

359.  A  Simple  Denominate  Number  is  a  number  expressed  in 
units  of  only  one  denomination. 

Thus,  3  yards,  and  2  days,  are  each  a  simple  denominate  number. 

360.  A  Compound  Denominate  Number  is  a  number  expressed 
in  units  of  more  than  one  denomination. 

Thus,  2  feet  6  inches,  4  days  6  hours,  are  each  a  compound  denominate 
number. 

361.  Reduction  of  Denominate  Numbers  is  the  process  of 
changing  them  to  equivalent  numbers  of  a  different  denomi- 
nation. 

362.  Reduction  Descending  is  the  process  of  changing  a 
number  to  an  equivalent  number  expressed  in  units  of  a  lower 
denomination. 

363.  Reduction  Ascending  is  the  process  of  changing  a 
number  to  an  equivalent  number  expressed  in  units  of  higher 
denominations. 

364.  Principle. — BeducMon  descendinc/  is  performed  by  mul- 
tiplication, and  reduction  ascending  is  performed  by  division. 

13* 


160  COMPOUND  NUMBERS. 

cj^se;  I. 
Reduction  Desceudiug-. 

365. — Ex.  1.  How  many  quarts  in  3  pecks  7  quarts?  In  3 
pecks  5  quarts  ?     In  1  bushel  1  peck  ? 

2.  How  many  inches  in  3  yards  2  feet  ?     In  4  yards  1  foot  ? 

3.  How  many  pennyweights  in  5  ounces  11  penny^veights  ? 
In  4  ounces  15  pennyweights  ? 

4.  How  many  days  are  9  weeks  5  days  ? 

5.  How  many  inches  in  ^  of  a  yard  ? 

Solution.  —  Since  in  1  yard  there  are  3  feet,  in  |  of  a  yard  there 
must  be  f  of  3  feet,  or  ^^  of  1  foot,  whicli  equals  f  of  a  foot.  And  since 
in  1  foot  there  are  12  inches,  in  |  of  a  foot  there  must  be  f  of  12  inches, 
or  20  inches.     Hence  f  of  a  yard  are  20  inches. 

6.  How  many  pounds  in  |  of  a  ton  ?     In  -J-j^  of  a  ton  ? 

7.  What  part  of  a  quart  is  -|-  of  a  gallon  ?     f  of  a  gallon  ? 

WRITTEN  EXEBCISKS. 

366. — Ex.  1.  How  many  pints  are  13  gal.  2  qt.  1  pt. 

13  gal.  2  qt.  1  pt. 
^  SoLrTioN. — Since  1  gallon  is  4 

■^rz"  .      -» o       7  quarts,   13  gallons    must    be    13 

oz  ISO.  qt.  in  lo  qal.  .-a         .         -o        *         a 

3  ^  times  4  quarts,  or  o2  quarts ;  and 

-^  52  quarts  +  2  quarts  are  54  quarts. 

54  No.  qt.  in  13  gal.  2  qt.               Since  1   quart    is  2   pints,   54 

Q  quarts  must  be  54  times  2  pints, 

or   108  pints;  and  108  pints -j-1 

108  No.  pt.  in  13  gal.  2  qt.            pj^t  are  109  pints.    Hence,  13  gal. 
1  2  qt.  1  pt.  are  109  pints. 

109  No.  pt.  in  13  gal.  2  qt.  1  pt. 

2.  How  many  yards  are  53  mi.  132  rd.  4  yd.  ? 

3.  How  many  pounds  are  5  T.  3  cwt.  15  lb.? 

367.  Rule  for  Reduction  Descenii\nq.— Multiply  the  number 
of  the  highest  dcnmnhiatioii  given,  by  that  number 
of  the  7iext  lower  which  equals  one  of  the  higher, 


COMPOUND   ^'^UMBEIiS.  151 

a7id  to  the  product  add  the  number,  if  any,  of  the 
Joiver  denomination. 

Reduce  this  result  in  like  manner,  and  so  proceed 
until  the  given  numher  is  reduced  to  the  required 
denomination. 


PnOBLJEMS. 

1.  How  many  square  yards  are  37  A.  132  sq.  rd.  ? 

2.  How  many  quarts  are  308  bu.  1  pk.  6  qt.  ? 

3.  How  many  quarters  are  iS%  yd.  1  qr.  ? 

4.  How  many  links  are  2  mi.  40  eh.  25  li.  ? 

5.  How  many  gills  in  16  hogsheads? 

6.  How  many  ounces  in  1  long  ton? 

7.  How  many  grains  are  7  oz.  19  pwt.  13  gr.  ? 

8.  How  many  square  rods  in  a  quarter  section  of  land  ? 

9.  How  many  square  rods  in  2  A.  5  sq.  eh.  8  sq.  rd.  ? 

10.  How  many  cubic  feet  are  20  cd.  6  cd.  ft.  ? 

11.  How  many  seconds  are  29  d.  12  h.  44  min.  3  sec.  ? 

12.  How  many  pints  are  f  of  a  bushel? 

9  o  a  SOLUTIOK.— Since   1   bushel   is   4 

-  hu.  =  jX4 pk.  =  -  ph.         pecks,  f  of  a  bushel  must  be  |  of  4 

pecks,  or  |  of  a  peck. 
-  p)k.  =  —  X.8qt.=-~  qt.  Since  1  peck  is  8  quarts,  |  of  a 

peck  must  be  |  of  8  quarts,  or  y  of 
J-  qt.  =  -r^X  2  pt.  =  -—  p^.         a  quart. 

J  Since   1   quart  is  2  pints,  ^-^   of  a 

-J-  pt.  =  i*5-  2^i'  quart  must  be  %*  of  2  pints,  or  ip 

of  a  pint  =  25f  pints. 


13.  How  many  pounds  are  y^g-  of  a  ton  ? 

14.  How  many  gills  are  f  of  a  hogshead? 

15.  What  part  of  a  second  is  yiroToTo-  ^^  ^  ^^^J  ^ 

16.  Reduce  .0525  cwt.  to  ounces. 

Solution. — .05^5  art.  =  .0525  X  100  Ih.  =  5.25  Ih.  =  5.25  X  16  oz.  =  84  oz. 

17.  Express  .09375  of  an  acre  in  square  rods. 

18.  Express  .7375  of  a  pound  Troy  in  penny^veights. 


152  COMPOUND   NUMBERS. 

19.  Reduce  |  of  a  rod  to  lower  integers. 

Solution.— Since  1  rod  is  5| 

jXS^  =^=3~  =  No.  of  yd.      -^^^'^''  '^  o^  ^  ^°^  ™"^^  be  f  of 
S         2        16  16  •>   ^  b\  yards,  or  S^V  yards. 

-^X3  =^  =  l^  =  No.  of  ft.  ^'^''  ^  rV'f/f '  "'^  '^  ^ 

^^  16  16  J  J  yard  must  be  y^  of  3  feet,  or  ly\ 

^  -^y  -17)      60       qS  ,1-       ,  .  feet. 

io  16  4  >>  bince  1  foot  is  12  inches. 


T? 


J  ,^  .  of   a   foot   must    be    y\   of   12 

-rd.=3yd.  Ift.S-in.  inches,  or  3f  inches.      Hence, 

f  of  a  rod  =  3  yd.  1  ft.  3|  in. 

20.  Reduce  ^  of  a  day  to  a  compound  number. 

21.  Express  f  of  a  hogshead  as  a  compound  number. 

22.  Express  y\  of  a  mile  as  a  compound  number. 

23.  Reduce  .7375  of  a  pound  Troy  to  a  compound  number. 

Solution.— .757J  lb.  =  .7375  X  12  oz.  =  8.85  oz. ,-  .85  oz.  =  .85  X  20  pict.  = 
17  pwt.     Hence,   .7375  oi  9.T^o\inA  =  8  oz.  17  jnct. 

24.  Reduce  .5625  of  a  day  to  a  compound  number. 

25.  How  many  pints  are  .015625  of  a  bushel? 

26.  Reduce  .7625  of  a  degree  to  lower  integers. 

27.  Express  the  value  of  3.076  cubic  yards  as  a  compound 
number. 

28.  Express  the  value  of  19.742  acres  as  a  compound 
number. 

cAsE  II. 

Reduction  Ascending. 

368. — Ex.  1.  How  many  pecks  in  31  quarts  ?  In  29  quarts? 
How  many  bushels  in  40  quarts  ? 

2.  How  many  yards  in  96  inches  ?     In  60  inches  ? 

3.  How  many  ounces  in  71  pennyweights?  In  63  penny- 
weights ? 

4.  How  many  yards  in  20  inches  ? 

Solution. — Since  12  inches  are  1  foot,  there  are  in  20  inches  as  many 
feet  as  y'j  of  20,  or  J  of  a  foot.  Since  3  feet  are  1  yard,  tliere  are  in  |  of 
a  foot  as  many  yards  as  \  of  f,  or  §  of  a  yard. 

5.  How  many  tons  in  14  hundred-weight? 

6.  How  many  gallons  in  5-  "f  «  quart? 


COMPOUND  NUMBERS.  163 

WMITTJEN   EXERCISES. 

i  369. — Ex.  1.  How  many  gallons  are  109  pints? 

2)109  Solution.— Since  2  pints  are  1 

,  quart,  there  must   be   one   half   as 

4.JOUf 1  pi.  many  quarts  as  pints,  or  54  quarts, 

13  gal.  .  .  2  at.  with  a  remainder  of  1  pint. 

Since  4  quarts  are  1  gallon,  there 
10  J  pt.  =  IJ  gal  ^  qt.  1  pt.       ^^^^  y^^  ^^^  foyj.jjj  ^j,  ^^^^^.  gaiio„s 

as  quarts,  or  13  gallons,  with  a  remainder  of  2  quarts.     Hence,  109  pt.  = 
13  gal.  2  qt.  1  pt. 

2.  How  many  miles  are  14400  rods? 

3.  How  many  tons  are  10315  pounds? 

370.  Ruie  for  Reduction  Ascending. — Divide  the  given 
number  by  that  number  of  its  rl enomination  jvhich 
equals  one  of  the  next  higher,  and  lujnte  tlie  remain- 
der, if  any. 

Divide  the  quotient  in  like  manner,  and  so  continue 
until  the  given  number  is  reduced  to  the  required  de- 
nomination. 

Tlie  last  quotient,  with  the  remainders ,  if  any, 
written  in  their  order  froin  the  highest  to  the  lowest, 
will  be  the  required  result. 

Reduction  Ascending  and  Reduction  Descending,  being 
performed  by  opposite  processes,  are  proofs  of  each  other. 

PItOBT.EMS. 

1.  How  many  acres  are  183073  square  yards? 

2.  How  many  bushels  are  9870  quarts? 

3.  How  many  yards  are  275  quarters  ? 

4.  How  many  miles  are  20025  links  ? 

5.  How  many  hogsheads  are  32256  gills  ? 

6.  How  many  tons  are  35840  ounces  ? 

7.  How  many  ounces  are  3829  grains  ? 

8.  How  many  quarter  sections  of  land  are  25600  square 
rods? 


154  COMPOUND  NUMBERS. 

9.  How  many  acres,  square  chains  and  square  rods  are  408 
square  rods  ? 

10.  How  many  cords  are  2556  cubic  feet? 

11.  How  many  days  are  2551443  seconds? 

12.  How  many  bushels  are  25f  pints  ? 

Solution. — Since  2  pints  are  1  quart, 
^g~  r=  —  —  No.  of  pt.       tliere  must  be  one  half  as  many  quarts 
2^g  ^  as  pints,  or  %^  of  a  quart. 

■z~-^ 2  =  -r~  ~ ^'^-    ^f  Q^-  Since  8  quarts  are  1  peck,  there  must 

g,  g  be  one  eightli  as  many  pecks  as  quarts, 

-y^  S  =  j=No.  of  2)k.       or  I  of  a  peck. 
S  ^  Since  4  pecks  are  1  bushel,  there  must 

5  ^'  -^^  S  ^^  ^J  be  one  fourth  as  many  bushels  as  pecks, 

or  f  of  a  bushel. 

13.  What  part  of  a  ton  is  375  pounds? 

14.  What  part  of  a  hogshead  is  1344  gills? 

1 5.  What  part  of  a  day  is  -^^  of  a  second  ? 

16.  What  decimal  of  a  hundred-weight  is  84  ounces? 

Solution.— 54  nz.  =  84  -^  16,  or  5.25  lb.  =  5.25  ~  100,  or  .0525  cwt. 

17.  What  decimal  of  an  acre  is  15  square  rods? 

18.  What  decimal  of  a  pound  Troy  is  177  pennpveights  ? 

19.  Reduce  3  yd.  1  ft.  3f  in.  to  a  fraction  of  a  rod. 

S~  tn.  =  ~~r  of  an  in. ;  --  ^  i^  =  -r ;  nence,  3-  m.  =  --  ft 

4  4      •'  4  16  '4  16  '' 

lift-=§  of  a  ft.;  jj*S=i:  /.e«ce,iji/(.  =  ^|   yd. 

3k'  y<^- = f »/ « !/*  '■  iF  f = I  '■  '--•  ■?i  y^- = I  '■''• 

Solution. — Since  12  inches  are  1  foot,  there  must  be  one  twelfth  as 
many  feet  as  inches,  or  y\  of  a  foot. 

Since  3  feet  are  1  yard,  there  must  be  one  third  as  many  yards  as  feet, 
or  -^-^  of  a  yard. 

Since  5^  yards,  or  V  of  a  yard,  arc  1  rod,  there  must  be  j-j-  as  ni;uiy 
rods  as  yards,  or  f  of  a  rod. 

20.  What  fraction  of  a  day  is  16  h.  36  min.  55y\  sec.  ? 

21.  Wliat  fraction  of  a  hogshead  is  39  gal.  8  pt.  3  gi.  ? 

22.  What  fraction  of  a  mile  is  85  rd.  1  yd.  2  ft.  6  in.  ? 


COMPOUND  NUMBERS.  155 

23.  Reduce  8  oz.  17  pwt.  io  a  decimal  of  a  pound. 

Solution.— i"i>-(?.  =  ir  -^  20,  or  .85  oz. ;  S.S5  oz.  =  S.SS  -^  12,  or  .7375  lb. 

24.  Reduce  13  h.  30  rain,  to  a  decimal  of  a  day. 

25.  Reduce  1  pint  to  a  decimal  of  a  bushel. 

26.  Reduce  45'  45"  to  a  decimal  of  a  degree. 

27.  Express  3  cu.  yd.  2  cu.  ft.  89.856  cu.  in.  as  a  mixed  deci- 
mal of  a  cubic  yard. 

28.  Express  19  A.  118  sq.  rd.  21.78  sq.  yd.  as  a  mixed  deci- 
mal of  an  acre. 

CJ^SE   III. 
One  Compound  Number  Reduced  to  the  Fraction  of  Another. 

371. — Ex.  1.  How  many  feet  are  3  yards  2  feet?     Are  5 
yards  1  foot? 

2.  What  fraction  of  16  feet  is  1  foot?     Is  11  feet? 

3.  Reduce  3  yards  2  feet  to  a  fraction  of  5  yards  1  foot. 

4.  What   fraction   of  4  pounds   3   ounces  is   2   pounds  5 
ounces  ? 

WBITTEIf    EXEBCISES. 

372.— Ex.  1.  Reduce  3  wk.  5  d.  to  a  fraction  of  11  wk.  5  d.  1  h. 

r,      ,      -   7         ^,T  /  7  Solution. — Since  only  similar 

3  wk.   O  d.  =  624-  fi-  r,  u  J    /  A  ^ 

^  numbers   can   be  compared    (Art. 

11  ivk.  5  d.   1  h.  :=  1969  h.      214),  we  reduce  each  of  the  given 

numbers  to  hours,  and  have  as  their 

equivalent  624  hours  and  19G9  hours. 

Since  1  hour  is  ^^^-^  of  1969  hours,  624  hours  must  be  y%W  of  1969 

hours.     Hence,  3  wk.  5  d.  are  j^^-^g  of  11  wk.  5  d.  1  h. 

2.  Reduce  5  mi.  40  rd.  to  a  decimal  of  8  mi.  20  rd. 

5  mi.  40  rd.  =  1640  rd. 

Solution. — 5  mi.  40  rd.  are  |f  f  g  = 

8  mi.  20  rd.  =  2580  rd.         j%%  of  8  mi.  20  rd. ;  and  T^y  =  .635  + . 

i/^  /n         oci  Hence,  5  mi.  40  rd.  are  .635  +of  8  mi. 

^^  =  -^=.^55+  20  rd. 

2580     129 

3.  What  fraction  of  6  gal.  1  qt.  1  pt.  is  2  gal.  0  qt.  1  pt.  ? 


156  COMPOUND    NUMBERS. 

373.  Rules  for  Reduction  of  one  Cempound  Number  to  the  Frac- 
tion of  another. -i.  Reduce  both  of  the  given  nuirvbers  to 
the  same  denomination  ;  and  then  make  the  nxvmher 
denoting  the  part  the  numerator-,  and  that  denoting 
the  whole  the  denominator  of  the  fraction  required. 

^.  When  the  fraction  required  is  a  decimal,  reduce 
the  comjyion  fractioji  thus  found  to  a  decimal. 

PItOBZEMS. 

1.  What  fraction  of  25°  42'  40"  is  7°  42'  48"? 

2.  "What  decimal  will  express  the  relation  of  5  cwt.  91  lb.  to 
2  T.  7  cwt.  28  lb.  ? 

3.  From  a  farm  containing  170  A.  16  sq.  rd.,  I  sold  37  A. 
128  sq.  rd.     What  part  of  the  farm  did  I  sell  ?  Ajis.  |. 

MISCELLANEOUS    PROBLEMS. 

374. — 1.  What  is  the  cost  of  .6725  of  a  hundred-weight 
of  butter,  at  40  cents  per  pound  ? 

2.  What  decimal  of  a  hundred-weight  of  butter,  at  40  cents 
per  pound,  can  be  bought  for  $26.90  ? 

3.  What  Avill  1  hogshead  4  gallons  1  quart  of  wine  cost,  at 
$5  per  gallon  ? 

4.  How  much  wine,  at  $5  per  gallon,  can  be  bought  for 
$336.25  ? 

5.  The  distance  between  two  places  on  the  same  parallel  of 
latitude  is  17°  30' ;  how  far  apart  are  they,  a  degree  in  that 
latitude  being  54  miles  ?  Ans.  945  miles. 

6.  A  boy  has  1  pk.  6  qt.  ^  pt.  of  chestnuts ;  what  part  of  a 
bushel  has  he  ? 

7.  How  many  ounces  of  gold  weigh  as  much  as  4  pounds  of 
lead?  Ans.  b^. 

8.  What  decimal  of  a  ton  of  nails,  at  5  cents  a  pound,  can 
be  bought  for  $2.40  ? 

9.  A  grocer  has  8316  eggs  to  pack  in  11  boxes  ;  how  many 
dozen  must  he  pack  in  each  box?  Aiis.    63. 


COMPOUXD   NUMBERS.  .  157 

10.  What  will   23  A.  120  sq.  rd.  of  land  cost,  at  $.50  per 
square  rod  ?  Ans.  $1900. 

11.  How  many  sheets  of  paper  are  12  reams  5  quires  18  sheets? 

12.  What  number  of  silver   spoons,  each  weighing    1  oz. 
9  pwt.,  can  be  made  from  2  pounds  of  silver  ? 


TEST    QUESTIONS. 

375. — 1.  What  is  a  Number?  A  denominate  number?  A  simple 
denominate  number?     A  compound  denominate  number? 

2.  What  is  Reduction  ?  How  do  reductions  descending  and  ascending 
differ  ?    Wliich  is  performed  by  multiplication  ?    Which  by  division  ? 

3.  What  is  Cancellation  ?  Upon  what  principle  does  -cancellation 
depend?  (Art.  129  — 3.)  How  is  a  fraction  reduced  to  its  lowest  terms? 
Upon  what  principle  does  the  process  depend?    (Art.  162.) 

4.  In  what  two  ways  may  a  fraction  be  multiplied  by  an  integer? 
How  may  a  number  be  multiplied  by  a  fraction  ?     (Arts.  189-195.) 

5.  In  what  two  ways  may  a  fraction  be  divided  by  an  integer?  In 
what  two  ways  may  a  number  be  divided  by  a  fraction  ?    (Arts.  203-200.) 


SECTION    XXXVI. 
ABBITIOJf  OF  COMPOU^'D  LUMBERS. 

376.— Ex.  1.  What  is  the  sum  of  2  T.  15  cwt.  25  lb. ;  3  T. 
0  cwt.  64  lb. ;  and  7  cwt.  16  lb.  ? 

Solution. — Since  only  units   of  like  kind 

2  T.   15  Clot.  25  lb.     can  be  added  (Art.  46 — 1),  write  the  numbers  so 

3  0  64-  *'^^*'  ""Its  of  the  same  denomination  shall  stand 
,y            1^            in  the  same  column. 

Begin  at  the  right,  and  add  the  numbers  of 


b  i.     o  civt.      o  lb.       each   denomination   in   the   order    of  tlie   de- 
nominations. 

The  sum  of  the  pounds  is  105  lb.,  or  1  cwt.  5  lb.  Write  the  5  lb.  as  the 
pounds  of  the  sum,  and  add  the  1  cwt.  with  the  sum  of  hundred-weights. 

The  sum  of  the  hundred-weights  is  23  cwt.,  or  1  T.  3  cwt.  Write  the 
3  cwt.  as  the  hundred-weights  of  the  sum,  and  add  the  1  T.  with  the 
column  of  tons. 

The  sum  of  the  tons  is  6  T.,  which  we  write  as  the  tons  of  the  sum. 

Therefore,  6  T.  3  cwt.  5  lb.  is  the  sum  required. 
14 


158  COMPOUND   yVMBEBS. 

2.  What  is  the  sum  of  810  yd.  1  ft.  10  in. ;  617  yd.  2  ft. 
11  in. ;  85  yd.  2  ft.  8  in. ;  679  yd.  5  in. ;  and  6  yd.  3  in.  ? 

3.  What  is  the  sum  of  3  lb.  9  oz.  18  pwt.  11  gr. ;  1  lb.  4  oz. 

19  pwt.  20  gr.;  and  1  oz.  0  pwt.  23  gr.? 

377.  Rule  for  Addition  of  Compound  Numbers.— TFV'^^e  the 
numbers  so  that  units  of  the  same  denomination  shall 
stand  in  the  same  column. 

Begin  ivith  the  lowest  denomination,  and  add  the 
numbers  of  each  denomination  separately .  If  the 
sum  is  less  than  one  of  the  next  higher  denomina- 
tion, ivrite  it  as  a  part  of  the  required  result. 

If  the  sum  is  equal  to  or  exceeds  one  or  more  units 
of  the  next  higher  denomination,  write  the  excess _ 
if  any,  as  a  part  of  the  required  result,  and  add  the 
number  of  units  of  the  higher  denomination  with  the 
numbers  of  that  denomination. 

FR'OBZEMS. 

1.  What  is  the  sum  of  13  lb.  6  oz.  11  pwt.  9  gr. ;  1  lb.  4  oz. 
13  pwt.  20  gr. ;  and  1  oz.  0  pwt.  13  gr.  ? 

2.  Find  the  sum  of  4  gal.  1  qt.  1  pt.  1  gi. ;  4  gal.  0  qt.  1  pt. 
3  gi. ;  5  gal.  3  qt.  0  pt.  2  gi. ;  and  10  gal.  2  qt.  1  pt.  4  gi. 

3.  Find  the  sum  of  71°  9'  59.5";  20^  24'  18.4";  and  19°  30'  34". 

4.  Find  the  sura  of  46  bu.  2  pk.  6  qt.  1  pt. ;  43  bu.  2  pk.  2  qt. 
1  pt. ;  86  bu.  1  pk.  3  qt. ;  68  bu.  3  pk.  1  qt.  1  pt. ;  76  bu.  2  pk. 
3  qt. ;  and  69  bu.  2  pk.  1  qt.  1  pt.       Ans.  391  bu.  2  pk.  2  qt. 

5.  Find  the  sum  of  30  A.  120  sq.  rd. ;  42  A.  60  sq.  rd. ;  80  A. 

20  sq.  rd. ;  and  150  sq.  rd.  .4ns.  154  A.  30  sq.  rd. 

6.  What  is  the  sum  of  100  rd.  5  yd.  2  ft. ;  150  rd.  0  yd.  2  ft. ; 
and  105  rd.  3  yd.  1ft.? 

100  rd.  5  yd.    2jt. 

150         0           2  Solution.— Since  1  yd.  =  1  ft. 

105         3           1  6  in.,  we  may  substitute  this  vahie 

'                              'i  for  the  \  yd.,  and  thus  obtain,  as 

1  mi.  36  rd.  3z  yd.  3  ft.  .,„  expression  of  the  result,  1  mi. 

Or,  3G  rd.  4  yd.  0  ft.  G  in. 

1  vii.  36  rd.  4  yd.  0  ft.  6  in. 


COMPOUND  NUMBERS.  169 

7.  What  is  the  siim  of  2  A.  120  sq.  rd.  10  sq.  yd. ;  3  A.  0  sq. 
rd.  12  sq.  yd. ;  and  140  sq.  rd.  20  sq.  yd.  ? 
Ans.  6  A.  101  sq.  rd.  llf  sq.  yd. 
Or,  6  A.  101  sq.  rd.  11  sq.  yd.  6  sq.  ft.  108  sq.  in. 
.    8.  What  is  the  sum  of  66  y.  99  d.  8  h.  50  rain.  ;  9  y.  1  d.  2  h. 
57  min. ;  6  y.  70  d.  1  h. ;  and  5  h.  50  min.  ? 

9.  What  is  the  sum  of  13  cu.  yd.  8  cu.  ft.  1030  cu.  in.;  20 
cu.  yd.  11  cu.  ft.  903  cu.  in.;  and  107  cu.  yd.  11  cu.  ft.  1240 
cu.  in.  ?  Aiu.  141  cu.  yd.  4  cu.  ft.  1445  cu.  in. 

10.  Find  the  sum  of  |  of  a  mile  and  |^  of  a  rod. 

Solution.- Since  |  mi.  =  177  rd.  4  yd.    Oft.  10  in. 
and  i'>-d-=  4  2        6J 


I  mi.  +  I  rd.  ^178  rd.  3^  yd.  0  ft.  3~ 


in. 


Or,  178  rd.  3  yd.     1  ft.     9-  in. 

11.  What  is  the  sum  of  5.141  tons  and  .3218  of  a  ton,  ex- 
pressed as  a  compound  number  ? 

12.  What  is  the  sum  of  .005  of  a  common  year  and  f  of  a 
week  ?  Ans.  6  d.  0  h.  36  min. 

13.  What  is  the  sum  of  3  gal.  2  qt.  0  pt.  1.4gi. ;  |  of  a  gal- 
lon ;  and  .875  of  a  hogshead  ?  ) 

14.  I  have  in  one  range  of  wood  13  cd.  3  cd.  ft. ;  in  a  second, 
21  cd.  48  cu.  ft. ;  and  in  a  third,  42  cd.  4  cd.  ft.  8  cu.  ft.  How 
much  have  I  in  all  ?  Ans.  77  cd.  2^  cd.  ft. 

-  15.  A  ship  sailing  from  Boston,  in  latitude  42°  20'  north,  to 
Cape  Horn,  55°  58'  15"  south,  passes  through  how  many  de- 
grees of  latitude  ?  >'  0      ' 

16.  Washington  is  77°  2'  48"  of  longitude  west  of  Greenwich, 
and  the  extreme  west  point  of  Alaska  is  91°  14'  12"  west  of 
Washington.  What  is  the  longitude  of  that  point,  reckoned 
from  Greenwich?  Ans.  168°  17'  W. 

17.  What  is  the  sum  of  the  following  measurements  :  2  yd. 
2  ft.  7  in. ;.  71  yd. ;  3  yd.  1  ft.  11  in. ;  li  rd.  5  yd.  1  ft.  6  in. ; 
and  2  rd.  16  ft.  6  in.  ?  Ans.  8  rd.  0  yd.  0  ft.  9  in. 


160  COMPOUND  NUMBERS. 

SECTION    XXXVII. 

&JJBTEACTIOX  OF  COMPOIWD  XUMBERS. 
378.— Ex.  1.  From  17  bu.  2  pk.  6  qt.  take  8  bu.  3  pk.  4qt.  * 

f '^  7       (D    1    r>  Solution. — Since    only  units   of   the    same 

1/  bu.  ^pk.  6  gf.        Yn\A  can  be  subtracted  the  one  from  the  other 

o         o  4-  (Art.  57 — 1),  write  the  subtrahend  under  the 

8  bu  3  pk  2  at  minuend,  so  that  units  of  the  same  kind  shall 
stand  in  the  same  column. 

Begin  at  the  right,  and  subtract  the  units  of  each  denomination  of  the 
subtrahend  from  those  of  the  same  kind  in  the  minuend. 

4  qt.  from  6  qt.  leave  2  qt.,  which  is  the  difference  of  the  quarts. 

Since  3  pk.  cannot  be  taken  from  2  pk.,  take  1  bu.  from  tlie  17  bu., 
leaving  16  bu.,  and  add  it,  reduced  to  pecks,  to  the  2  pk.,  thus  obtaining 
6  pk. ;  then,  3  pk.  from  6  pk.  leave  3  pk.,  which  is  the  difference  of  the 
pecks. 

8  bu.  from  16  bu.  leave  8  bu.,  which  is  the  difference  of  the  bushels. 

Therefore,  8  bu.  3  pk.  2  qt.  is  the  difference  required. 

2.  From  35  lb.  14  oz.  take  19  lb.  15  oz.     Ans.  15  lb.  11  oz. 

379.  Rule  for  Subtraction  of  Compound  Numbers.—  Write  the 
siihtjxdiend  under  the  minuend ,  so  that  units  of  the 
same  denomination  shall  stand  in' the  same  column. 

Begin  with  the  loivest  denomination,  and  suhtract 
the  numher  of  units  of  each  denomination  of  the  sub- 
trahend from  the  numher  of  units  of  the  same  de- 
nomination of  the  minuend,  if  possihle,  and  write 
the  difference  beneath  as  apart  of  the  required  differ- 
ence. 

If  the  number  of  any  denomination  of  the  subtra- 
hejid  is  greater  than  that  above  it,  increase  the  upper 
number  by  adding  to  it  as  rjvany  units  as  are  one  of 
the  next  higher  denomination,  and,  subtract;  then, 
regarding  the  number  of  units  of  the  next  higher 
denomination  of  the  minuend  as  one  less,  proceed  as 
before. 


coif  FOUND  NUMBERS.  161 

PR  OBZ,  EMS. 

1.  From  12  cwt.  85  lb.  11  oz.  take  7  cwt.  58  lb.  6  oz. 

2.  From  1  lihd.  34  gal.  2  qt.  1  pt.  8  gi.  take  45  gal.  3  qt.  Ipt. 
2  gi.  Ans.  51  gal.  3  qt.  1  gi. 

3.  From  14  yd.  2  qr.  take  9  yd.  3  qr. 

4.  From  78°  55'  0"  take  71°  4'  20".  ^jis.  7°  50'  40". 

5.  From  116  cd.  4  cd.  ft.  6  cu.  ft.  1620  cu.  in.  take  105  cd. 
5  cd.  ft.  7  cu.  ft.  1511  cu.  in. 

6.  Subtract  7  lb.  7  oz.  10  pwt.  23  gr.  from  21  lb.  4  oz.  14  pwt. 
13  gr.  Ans.  13  lb.  9  oz.  3  pwt.  14  gr. 

7.  Subtract  5  mi.  215  rd.  5  yd.  from  8  mi.  216  rd.  3  yd. 

Atis.  3  mi.  3|-  yd.,  or  3  mi.  3  yd.  1  ft.  6  in. 

8.  Take  .0038  of  a  year  from  f  of  a  week. 

Solution.  -Since  |  wk.  =     ^  d.    19  h.     12  min.    0  sec. 
and  .0038  y.   =     1  9         17  16.8 

~iuh.  —  .0038y.   =    Id.      9h.    54  min.  43.2  sec. 
5  '' 

9.  Take  |^  of  a  square  yard  from  1  rd.  21  sq.  ft.  56  sq.  in. 

10.  Take  |  of  a  great  gross  from  9.125  gross. 

11.  From  1^  of  a  bushel  take  3  pk.  0  qt.  1  pt. 

12.  A  man  has  travelled  4  mi.  64  rd.  How  much  farther 
must  he  go  to  have  travelled  6  miles  ? 

.  13.  From  a  cask  containing  36  gal.  1  qt.  of  molasses,  19  gal. 
2  qt.  1  pt.  have  been  drawn.     How  much  remains  in  the  cask  ? 

DIFFEEENCE    OF    DATES. 

380. — In  computation  of  the  Difference  of  Dates,  centuries 
are  numbered  from  the  beginning  of  the  Christian  era,  the 
months  from  the  beginning  of  the  year,  and  the  days  from 
the  beginning  of  the  month. 

Thus,  May  23,  1871,  is  the  23d  day  of  the  5th  month  of  the  71st  year 
of  the  19th  century. 

In  estimating  the  difference  between  dates,  the  entire  calendar  montlis 
are  found,  and  the  remaining  days  counted. 

Any  number  of  days  less  than  30,  in  business  transactions,  are  usually 
regarded  as  the  same  number  of  thirtieths  of  a  month. 
14  » 


It32  COMPOUND  NUMBERS. 

381. — Ex.  1.  A  man  left  home  on  a  journey,  July  17,  1867, 
and  returned  November  12,  1869.    How  long  was  he  absent '( 

Yroiry  Solution. 

1869  y.    11  mo.   12  d.  July  17, 1867,  to  July  17, 1869  =  2  y. 

1867  7  17  "  July  17, 1869,  to  Oct.  17,  1869  =  3  mo. 

"  Oct.  17, 1869,  to  Nov.  12, 1869  =  26  d. 


2  ^J.     3  mo.    26  d.    fience,  the  entire  difference  is  2  y.  3  mo.  26  d. 

2.  A  man  was  born  May  16,  1819;  how  old  was  he  Septem- 
ber 23,  1862  ?  Ans.  43  y.  4  mo.  7  d. 

3.  If  a  note  dated  February  25,.  1868,  Avas  paid  July  11, 
1869,  how  long  did  it  remain  unpaid  ?    Ans.  1  y.  4  mo.  16  d. 

4.  The  late  civil  war,  which  continued  4  y.  1  mo.  14  da.,  ter- 
minated May  26,  1865.    When  did  it  begin  ? 

Ans.  April  12,  1861. 

5.  The  American  Revolution  began  April  19, 1775,  and  ter- 
minated January  20,  1783.    How  many  years  did  it  continue? 


SECTION    XXXVIII. 

MULTIPLICATIOJr  OF  COMFOVXB  JfUMBERS. 

382— Ex.  1.  Multiply  6 gal.  3 qt.  1  pt.  by  5. 

6  gal.   3  qt.    1  pt.  Solution. — Write  the  multiplier  under  the 

/-  lowest  denomination  of  the  multiplicand,  and, 

beginning  at  the  right,  multiply  the  number 


34-  gal.  1  qt.  1  pt.  of  each  denomination  in  the  order  of  the  de- 
nominations. 

Five  times  1  pt.  are  5  pt.,  or  2  qt.  1  pt.  Write  the  1  pt.  as  the  number 
of  that  denomination  in  the  product,  and  reserve  the  2  qt.  to  be  added  to 
the  product  of  the  quarts. 

Five  times  3  qt.  are  15  qt. ;  15  qt.  and  2  qt.  are  17  qt.,  or  4  gal.  1  qt. 
Write  the  1  qt.  as  the  number  of  that  denomination  in  the  product,  and 
reserve  the  4  gal.  to  be  added  to  the  product  of  the  gallons. 

Five  times  6  gal.  arc  30  gal. ;  30  gal.  and  4  gal.  are  34  gal.,  which  write 
as  the  gallons  of  the  product. 

The  entire  product  is  34  gal.  1  qt.  1  pt. 

2.  Multiiily  12  bu.  3  pk.  1  qt.  by  7. 

3.  Multiply  7  yd.  3|  qr.  by  8.  Ans.  63  yd.  2  qr. 


COMPOUND    NUMBERS.  163 

383.  Rule  for  Multiplication  of  Compounii  Numbers.— TJ^V-j^e  the 
multiplier  luider  the  lowest  cleuoDii nation  of  the 
multiplicand. 

Begin  udth  the  lowest  denomination,  and  multiply 
the  juvmber  of  each  denomination  in  its  order .  If  the 
product  is  less  than  one  of  the  next  higher  denomina- 
tion, write  it  as  a  part  of  the  required  product. 

If  tlie  product  is  equal  to  or  exceeds  one  oi^  more 
units  of  the  next  higher  denomination,  ivrite  the  ex- 
cess, if  any,  as  a  part  of  the  required  product,  and 
add  the  number  of  units  of  the  next  higher  denomi- 
nation to  the  product  of  that  denomination. 

PROBLE  3rS. 

1.  Multiply  16°  58'  26f"  by  9.  Ans.  152=  46'  2". 

2.  Multiply  15  lb.  5  oz.  13  pwt.  by  11. 

3.  Multiply  2  gal.  1  qt.  1  pt.  2  gi.  by  19. 

Ans.  46  gal.  1  qt.  0  pt.  2  gi. 

4.  Multiply  1  T.  17  cwt.  92  lb.  by  28. 

5.  One  ship  is  in  5°  15'  45"  north  latitude,  and  another  is  5 
times  as  far  north.     Wha.t  is  the  latitude  of  the  latter  ? 

Am.  26°  18'  45"  north. 

6.  If  a  team  can  draw  in  one  load  1  cd.  1|-  cd.  ft.  of  wood, 
how  much  can  it  draw  in  14  loads  ? 

7.  I  bought  4  packages  of  medicine,  each  containing  3  lb. 
4s6ol9  16gr.     What  is  the  weight  of  the  whole  ? 

Ans.  131b.  7^  23  19  4gr. 

8.  A  farm  consists  of  9  fields,  each  containing  12  A.  72  sq.  rd. 
What  is  the  extent  of  the  farm  ? 

9.  If  a  steamer  sail  211  mi.  192  rd.  a  day,  hoAv  far  will  it  sail 
in  15  days? 

10.  How  much  time  in  100  years,  each  365  d.  5  h.  48  min. 
49.7  sec.  long  ?  Ans.  36524  d.  5  h.  22  min   50  sec. 

11.  A  lot  of  laud  is  divided  into  6  house-lots,  eac  i  of  which 
contains  1  A.  4  sq.  rd.  120  sq.ft.  How  much  land  is  there  in 
all  the  lots?  A7is.  6  A.  26  sq.  rd.  175|-  sq.  ft. 


164  COMPOUSD  NUMBERS. 

SECTION    XXXIX. 

BIYISIOX  OF  COMPOUJS'D  J\^UMBERS. 

384.— Ex.  1.  Divide  34  gal.  1  qt.  1  pt.  by  5. 

5)34  gal.  1  qt.  1  Jit.  Solution.— Write  the  divisor  at  the  left  of 

~T>       J  Q    i    1  ^^®  dividend,  and  beginning  at  the  left,  divide 

ga  .      q  .      jj  .       ^^q  number  of  each  denomination  in  its  order. 

One  fifth  of  34  gal.  is  6  gal.,  with  a  remainder  of  4  gal.  Write  the 
6  gal.  as  the  gallons  of  the  quotient;  the  4  gal.  =  16  qt.,  which  added 
to  the  1  qt.  =  17  qt. 

One  fifth  of  17  qt.  is  3  qt.,  with  a  remainder  of  2  qt.  Write  the  3  qt, 
a-s  the  quarts  of  the  quotient ;  the  2  qt.  are  4  pt.,  which,  added  to  the 
1  pt.,  =  5  pt. 

One  fifth  of  5  pt.  is  1  pt.,  which  write  as  a  part  of  the  quotient. 

The  entire  quotient  is  6  gal.  3  qt.  1  pt. 

2.  Divide  89  bu.  1  pk.  7  qt.  by  7.      A7is.  12  bu.  3  pk.  1  qt. 

3.  Divide  39  lb.  7  oz.  8  pwt.  9  gr.  by  6. 

A71S.  6  lb.  7  oz.  4  pvrt.  17|^gr. 

385.  Rule  for  Division  of  Compound  Uumhers.  — Be£iji7i7'7?d 
with  the  highest  denomination,  divide  the  mvnvber  of 
each  denomination  in  its  order,  and  write  the  several 
quotients  as  thepaHs  of  the  same  denominations  of  the 
required  quotient. 

If  there  are  paHial  remainders,  reduce  each  to  the 
next  lower  denomination,  and  add  the  same  to  the 
number  of  that  denomination  before  dividing  it. 

When  divisor  and  dividend  are  both  compound 
numbers,  they  must  be  reduced,  to  simple  denominate 
numbers  of  the  same  denomination  before  dividing. 

V  no  Til,  KMS  . 

1 .  DivMe  67  yd.  2  qr.  liy  8.  Am.  8  yd.  If  qr. 

2.  Diviie  23  cu.  yd.  0  cu.  ft.  12  cu.  in.  by  4. 

Ans.  5  cu.  yd.  20  cu.  ft.  435  cu.  in. 

3.  Divide  53  T.  1  cwt.  76  lb.  by  28. 


COMPOUND  NUMBERS.  165 

4.  In  9  equal  lots  of  land,  taken  together,  there  are  112  A. 
8  sq.  rd.     What  is  the  extent  of  each  lot  ? 

Ans.  12  A.  72  sq.  rd. 

5.  A  quantity  of  tea,  consisting  of  19  equal  parcels,  con- 
tains 3  cwt.  32  lb.  8  oz.    What  is  the  weight  of  a  single  parcel  ? 

Ans.  17  lb.  8  oz. 

6.  Divide  30°  2'  by  2°  30'  10". 

S0°  2'  =  108120"  Solution. — When,  as  in  this  prob- 

lem, the  divisor  and  dividend  are  simi- 
2     oO    10     =^9010  lar  compound  numbers,  reduce  both  to 

mo-io    I'        an  in"        -i a      the  lowest  denomination  mentioned  in 
lUai^U      ■    JUIU  1^      either,  and  divide  as  in  simple  numbers. 

7.  How  many  kegs,  each  containing  6  gal.  3  qt.  1  pt.,  can  be 
filled  from  a  cask  containing  34  gal.  1  qt.  1  pt.  ? 

8.  If  8  yd.  If  qr.  are  required  for  a  suit  of  clothes,  how  many 
suits  can  be  made  from  67  yd.  2  qr.  ? 

9.  How  many  loads,  of  IT.  17  cwt.  92  lb.  each,  are  there  in 
53  T.  1  cwt.  76 \b.  of  hay?  Ans.  28. 

LONGITUDE    AND   TIME. 

386. — The  earth  turns  on  its  axis  from  west  to  east  once  in 
24  hours.  This  causes  the  sun  to  appear  to  pass  around  the 
earth  from  east  to  west  in  the  same  time. 

The  sun  appears  sooner  to  places  east  of  any  given  point  on 
the  earth  than  to  those  west  of  it.  Hence,  of  any  two  given 
places,  the  one  farthest  east  has  the  later  time,  and  the  one 
farthest  west  the  earlier  time. 

Since  the  cirdlimference  of  any  circle  is  360°,  the  sun  ap- 
pears to  pass  over  ^  of  360°  of  the  earth's  circumference,  or 
15°  of  longitude,  in  1  hour ;  -^  of  15°,  or  15',  in  1  minute  °. 
and  g^o  of  15',  or  15",  in  1  second.     Hence  the  following 

COMPARISON    OF   LONGITUDE   AND   TIME. 

15°  of  Longitude  correspond  to  1  hour  in  time. 
15'  of  Longitude  "  1  ininute  in  time. 

15"  of  Longitude  "  1  second  in  time. 


168  COMPOUND  NVMBERS. 

387. — Ex.  1.  The  difference  in  longitude  between  Washing- 
ton and  San  Francisco  is  45°  23'  27" ;  what  is  the  difference 
in  time  ? 

15)  45°    23'         27"  Solution.— Since  15°  of  difference 

^  ,       ^      .       crxD-i  in  longitude  correspond  to  1  hour  dif- 

o  h.   Imxn.  oo-- sec.      r.  .    ^.       ,.,  ,.„,  .    -, 

5  terence  in  time,  lo  dirierence  m  longi- 

tude correspond  to  1  minute  difference  in  time,  and  \r>"  difference  in 
longitude  correspond  to  1  second  difference  in  time,  45°  23'  27^''  of  dif- 
ference in  longitude  must  correspond  to  jJ-  as  many  hours,  minutes  and 
seconds  respectively,  or  to  3  h.  1  min.  33f  sec. 

•  2.  The  difference  in  time  between  Washington  and  San 
Francisco  is  3  h.  1  min.  33f  sec. ;  what  is  the  difference  in 
longitude  ? 

3  h.      1  min.    33^  sec.  Solution. — Since  1  second  difference 

^  in  time  corresponds  to  15'^  difference 

^  in  longitude,   1  minute   difference   in 

45°       23'      27"  time  to  15'  of  difference  in  longitude, 

and  1  hour  difference  in  time  to  15° 
difference  in  longitude,  3  h.  1  min.  33f  sec.  difference  in  time  must  cor- 
respond to  15  times  as  many  seconds,  minutes  and  degrees  of  longitude 
respectively,  or  to  45°  23'  27". 

388.  Rules  for  Longitude  and  Time.— i.  Divide  the  dijfer- 
ence  of  longitude  hy  lo,  and  the  ninnher  of  degrees, 
minutes  and  seconds  of  the  quotient,  respectively ,  will 
he  the  hours,  minutes  and  seconds  of  the  difference 
of  time. 

2.  Multiply  the  difference  of  time  by  15,  and  the 
number  of  the  seconds,  minutes  and  hours  of  the 
product,  respectively,  will  he  the  seconds,  minutes 
and  degrees  of  longitude. 

PltOBLJE^rS. 

1.  When  it  is  12  o'clock  M.  at  that  part  of  Alaska  which 
is  87°  14'  30"  west  of  Boston,  what  is  tlie  time  in  Boston  ? 
Ans.  48  min.  58  sec.  past  5  o'clock  p.m. 


COMPOUND  NUMBERS.  167 

2.  The  time  at  Philadelphia  is  5  h.  0  min.  40  sec.  earlier  than 
that  of  Greenwich  ;  what  is  the  longitude  of  Philadelphia  ? 

Ans.  15°  10'  W. 

3.  The  difference  in  time  between  Portland  and  Chicago  is 
1  h.  9  min.  25\^^  sec. ;  what  is  the  difference  in  longitude  ? 

A71S.  17°  21'  26". 

4.  When  it  is  midnight  at  Canton,  113°  15'  east,  what  time 
is  it  at  New  Orleans,  90°  7'  west? 

Ans.  26  min.  32  sec.  after  10  o'clock  a.  m. 


TEST   QUESTIONS. 

389. — 1.  In  what  respects  is  the  Addition  of  compound  numbers  like 
addition  of  simple  numbers?  Why  cannot  dissimilar  denominate  num- 
bers be  added  ? 

2.  What  is  the  difference  between  Subtraction  of  simple  and  com- 
pound numbers  ?     How  may  the  subtraction  be  proved  ? 

3.  From  what  are  the  centuries  numbered?  Months?  Days?  What 
is  the  process  of  finding  the  diflerence  between  dates  ? 

4.  In  what  respects  is  Multiplication  of  compound  numbers  like 
multiplication  of  simple  numbers?  Why  cannot  a  compound  number 
be  multiplied  by  a  denominate  number?  (Art.  71 — 1.) 

5.  When  in  Division  of  compound  numbers  the  divisor  is  an  abstract 
number,  what  kind  of  a  number  will  the  quotient  be?  When  the 
divisor  and  dividend  are  similar  compound  numbers,  what  must  be  done 
before  dividing  ?     What  kind  of  a  number  will  the  quotient  be  ? 

6.  Why  does  the  sun  appear  to  pass  around  the  earth  from  east  to 
west  ?  Of  two  places  on  the  earth  having  different  longitude,  which  has 
the  earlier  time  ? 

7.  What  part  of  the  earth's  circumference  does  the  sun  appear  to  pass 
over  in  1  hour?  To  what  in  time  do  15°  of  longitude  correspond? 
15^  of  longitude?  15'^  of  longitude?  What  are  the  rules  for  longitude 
and  time? 

8.  What  is  the  Rule  for  addition  of  simple  numbers?  (Art.  49.) 
The  rule  for  addition  of  compound  numbers?  For  subtraction  of  simple 
liumbers?  (Art,  60.)     For  subtraction  of  compound  numbers? 

9.  What  is  the  rule  for  multiplication  of  simple  numbers?  (Art.  75.) 
For  the  multiplication  of  compound  numbers?  For  the  division  of 
simple  numbers?  (Art.  91.)     For  the  division  of  compound  numbers? 


168  ANALYSIS. 

SECTION    XL. 
ANALYSIS  BY  ALIQUOT  PARTS. 

390. — Ex.  1.  How  many  hundred- weight  is  one  half  of  a 
ton  ?  One  fourth  of  a  ton  ?  One  fifth  of  a  ton  ?  One  eighth 
of  a  ton  ?     One  tenth  of  a  ton  ? 

2.  What  part  of  an  acre  is  80  square  rods?  Is  40  square 
rods  ?     32  square  rods  ?     20  square  rods  ? 

3.  What  part  of  a  year  is  6  months?  Is  4  months?  3 
months?  2  months?  What  part  of  a  month  is  15  days?  Is 
10  days?     3  days? 

4.  What  will  one  half  a  ton  of  hay  cost,  at  $22  j^er  ton  ? 
One  fourth  of  a  ton,  at  $24  per  ton  ? 

5.  How  much  will  2  A.  80  sq.  rd.  of  land  cost,  at  $30  per 
acre  ?     5  A.  40  sq.  rd.,  at  $20  per  acre  ? 

6.  How  much  will  it  cost  a  man  for  2  mo.  3  d.  board,  at  $20 
per  month  ? 

DEFINITION". 

Analysis  by  Aliquot  Parts,  or  Practice,  is  a  concise  method 
of  computation  by  employing  aliquot  parts.   (Art.  279.) 

WniTTEN  EXERCISES. 

391. — 1.  What  is  the  rent  of  a  store  for  1  year  5  months  10 
days,  at  $300  a  year  ? 

Solution. 

The  rent  for  1  y.  is  $300.00 

4  mo.  "j  of  $300=      100.00 

Imo.  "  ^  of    100=         25.00 

lOd.  "  I  of      25=  8.33J 

"      ly.  5  mo.  10  d.  "  $433.33 j 

2.  At  $80  a  ton,  wliat  will  5  T.  15  cwt.  50  lb.  of  iron  cost? 

Ans.  $462. 


AXALYNZS.  169 

3.  At  S.96  per  gallon,  what  will  3  gal.  2  qt.  1  pt.  of  molasses 
cost? 

4.  How  much  will  9  months  24  days  of  labor  amount  to,  at 
$600  a  year  ?  Ans.  S490. 

5.  What  is  the  cost  of  constructing  20  mi.  120  rd.  of  road,  at 
84000  per  mile?  Ans.  $81500. 

6.  At  $6  a  hundred-weight,  what  must  be  paid  for  137;"» 
pounds  of  fish?  ^?js.  $82.50. 

7.  What  is  the  cost  of  360  yards  of  camlet,  at  $.621  per 
yard  ? 

Solution. 

At  $.50  a  yard,  the  coat  of  P>GO  yd.  is  |  of  $360-^$180.00 
"     .12^       "         "  380yd."  j  of     180^      45.00 

"$.62^       "         "  360  yd.''  $225.00 

$.62^  =  i  of  $1  +  i  of  ^  of  $1.  At  i  of  a  dollar  a  yard  the  cost  of 
360  yards  is  \  of  $360,  or  $180.  At  \  of  |  of  $1  a  yard  the  cost  is  J  of 
$180,  or  $45.     Hence,  at  $.62^  a  yard,  the  cost  is  $180  +  $45,  or  $225. 

8.  What  is  the  cost  of  180  bushels  of  corn,  at  $1.12|  per 
bushel  ? 

9.  What  is  the  cost  of  460  yards  of  cloth,  at  $5.75  per 
yard  ? 

10.  What  is  the  cost  of  172  bushels  of  rye,  at  $.87^  per 

bushel  ? 

Solution. 

At  $1.00 per  hu.,  the  cost  of  172 bit.  is  $172. 00 

«       .12-     "  "  172hu.''~of  $172=     21.50 

"$  .87J     "  "  172  bu."  $150.50 

Since  $.87^  is  \  of  a  dollar  less  than  a  dollar,  at  $.87i^  per  bu.  the  cost 
of  172  bu.  will  be  \  of  $172  less  than  $172,  or  $150.50. 

11.  What  is  the  cost  of  1671  poimds  of  tea,  at  $.83^  per 
pound  ?  Ans.  $1392.50. 

12.  What  is  the  cost  of  4  T.  16  cwt.  20  lb.  of  hay,  at  $25  per 
ton?  tItis.  $120.25. 

15 


170 


RECTANGULAR    MEASUREMENTS. 


SECTION    XLI. 
BECTAJfGULAR  MEASUKEMEJ^TS. 

Cj^SE   I. 

Surfaces. 

392. — Ex.  1.  How  many  square  inches  are  there  in  a  rec- 
tangular surface  which  is  15  inches  long  and  1  inch  wide  ? 

2.  At  3  cents  per  square  foot,  how  much  must  be  paid  for  a 
rectangular  board  which  is  25  feet  long  and  1  foot  broad  ? 

3.  How  many  square  rods  are  there  in  a  walk  which  is  50 
rods  long  and  1  rod  wide  ? 


DEFINITIONS. 

393.  The  Dimensions  of  a  rectangular  surface  are  the  length 
and  breadth,  or  width,  of  that  surface. 

394.  The  Unit  of  Measure  for  surfaces  is  always  a  square 
whose  dimensions  are  known ;  as  1  square  inch,  1  square  foot, 
etc. 

395.  The  Area  of  a  surface  is  the  number  of  times  the  sur- 
face contains  a  given  unit  of  measure. 

Thus,  the  rectangle  in  the  margin  will  be  seen 
to  contain  12  square  inches,  if  it  be  supposed 
to  be  4  inches  long  and  3  inches  wide. 

For,  upon  each  inch  of  length  there  may  be 
conceived  to  be  1  square  inch,  making  a  row 
of  4  square  inches,  and  as  tliere  will  be  as  many 
such  rows  as  there  are  inches  in  the  width,  or 

3  rows,  the  area  of  the  rectangle  must  be  3  times 

4  square  inches,  or  12  square  inches. 


306.  Principle. —  The  area  of  a  rectangle  is  equal  to  the 
number  of  square  vnifs  denotcil  by  the  product  of  the  number  of 
linear  xinits  in  the  Irnr/th,  multiplied  by  the  mimber  of  the  same 
linear  units  in  the  -width,  the  square  units  having  the  same  name 
as  the  linear  units. 


RECTAJS^GVLAR  MEASUREMENTS.  171 

WJtITTX:N  EXERCISES. 

307. — Ex.  1.  How  many  square  feet  of  surface  has  a  rec- 
tangular table  whose  length  is  7  feet  5  inches  and  width  5  feet 
4  inches  ? 

89X64  =  5696  No.  of  sq.  in.  Solution.— 7  ft.  5  in.  ==  89 

5696  sq.  in.  -  39  sq.  ft.  80  sq.  in.     !""^^'''  ^"'^  ^  ^^-  ^  '''■  =  ^^ 

The  product  of  89  by  64,  or  5696,  must  denote  the  number  of  square 
inches  of  surface  ;  and  5696  square  inches  are  equal  to  39  sq.  ft.  80  sq.  in., 
which  is  the  surface  required. 

2.  In  a  floor  16  feet  long  and  11  feet  wide,  are  how  many 
square  feet?  Ans.  176. 

3.  The  area  of  a  rectangular  floor  is  176  square  feet,  and  its 
length  is  16  feet.     What  is  its  width  ? 

]^'^0  H-  XO  =  H   ft.  Solution.— Since  the  product  of  the  num- 

ber of  linear  units  in  the  length  by  the  number 
in  the  width  is  equal  to  the  number  of  square  units  in  the  area,  the 
number  of  linear  units  in  the  required  dimension  must  equal  the  quotient 
of  the  number  of  the  square  units  of  the  area  divided  by  the  number  of 
linear  units  in  the  given  dimension,  or  11  feet. 

4.  The  area  of  a  board  is  45  square  feet,  and  its  width  1^ 
feet.     What  is  its  length  ? 

398.  Rulesfor  Measurements  of  Rectangular  Surfaces.— i,  Mul- 
tiply  the  length  by  the  width,  and  the  product  will 
denote  the  area. 

2.  Divide  the  area  by  either  of  the  dimensions,  and 
the  quotient  will  denote  the  other  dimension. 

PROBZESrS. 

1.  How  many  acres  are  there  in  a  rectangular  field  whose 
length  is  80  rd.  and  width  20  rd.  ?  Ans.  10. 

2.  The  area  of  a  field  is  4608  sq.  rd.,  and  its  Avidth  is  16  rd. ; 
what  is  its  length  ? 

3.  How  many  square  yards  of  carpeting  will  cover  a  room 
13|  ft.  square  ?  Ans.  20|. 


172 


RECTANGULAR  MEASUREMENTS. 


4.  A  i^ath  is  18  ft.  8  in.  long,  and  5  ft.  3  in.  wide.     What  is 
its  area  in  square  feet  ? 

Solution.— 18  ft.  8  in.  =  18|  ft.  =  \^  ft.,  and  5  ft.  3  in.  =  5^  ft.  =  ^  ft. 
V  X  V  ^^  9^-     ilence,  the  area  required  is  98  sq.  ft. 

5.  If  it  take  32^  sq.  yd.  of  carpeting  to  cover  a  floor  whose 
width  is  14  ft.,  what  is  the  length  of  the  floor  ? 


CA.se  II. 
Solids. 

399. — Ex.  1.  How  many  cubic  feet  are  there  in  a  rectangu- 
lar beam  whose  length  is  20  ft.,  width  1  ft.  and  thickness  1  ft.  ? 

2.  What  is  the  value  of  a  stick  of  timber  12  ft.  long,  1  ft. 
wide  and  1  ft.  thick,  at  10  cents  per  cubic  foot? 


DEFINITIONS. 

400.  The  Dimen.sions  of  a  rectangular  solid,  or  volume 
(Art.  319),  are  the  length,  breadth  or  width,  and  thickness, 
depth  or  height. 

401.  The  Unit  of  Measiii'e  for  solids,  or  volumes,  is  always 
some  cube  (Art.  317)  whose  dimensions  are  known  ;  as,  1  cubic 
inch,  1  cubic  foot,  etc, 

402.  The  Cubic  Contents,  or  capacity, 
of  a  solid,  or  volume,  is  the  number  of 
times  the  solid  or  volume  contains  a 
given  unit  of  measure. 

Thus,  tlie  rectangular  solid,  or  volume,  in 
the  margin  will  be  seen  to  contain  72  cubic 
inches,  if  it  be  supposed  to  be  6  inches  long, 
3  inches  wide  and  4  inches  thick. 

For,  upon  each  of  the  18  square  inches  of  the  lower  fiice  there  m;iy 
bo  conceived  to  be  1  cubic  inch,  making  a 
layer  of  18  cubic  inches  ;  and  as  there  will  be 
as  many  such  layers  as  there  are  inches  of 
thickness,  or  4,  the  contents  of  the  volume 
must  be  4  times  18  cubic  inches,  or  72  cubic 
iuche». 


yyy  .^ 


7^7 


Z17 


RECTANGULAR   MEASUREMENTS.  173 

403.  Principle. —  The  cubic  contents  of  a  solid,  or  volume, 
are  equal  to  the  number  of  cubic  units  denoted  by  the  product  of 
the  number  of  the  same  linear  units  in  the  length,  width  and 
thickness,  the  cubic  units  having  the  same  name  as  the  linear  units. 


WRITTEN  EXERCISES. 

40-1:. — Ex.  1.  How  many  cubic  feet  are  there  in  a  rectangular 
block  of  marble  which  is  8  ft.  long,  3  ft.  6  in.  wide  and  2  ft. 
3  in.  thick  ? 

8 

3^  Solution.— 3  ft.  6  in.  =  3^  ft.,  and  2  ft.  3  in.  =  2\  ft. 


28 


The  product  of  the  number  of  units  in  the  length, 
width  and  thickness  is  63,  which  must  be  the  number  of 
2—  cubic  feet  required. 

63  m.ft. 

2.  What  are  the  cubic  contents  of  a  body  20  ft.  long,  6  ft. 
Avide  and  4  ft.  thick  ? 

3.  A  block,  containing  15625  cubic  inches,  is  2  feet  1  inch 
wide  and  2  feet  1  inch  thick.     What  is  its  length  ? 

Aiis.  2  ft.  1  in. 

4.  A  rectangular  body  whose  cubic  contents  are  480  cu.  ft. 
is  20  ft.  long  and  6  ft.  wide.     What  is  its  thickness  ? 

SO  X  6"  =  120  Solution. — Since  the  number  of  cubic  feet  of 

contents  must  be  the  product  of  tlie  number  of 

120)4-80  units  in  the  length,  width  and  thickness,  the  quo- 

/  tient  of  480  divided  by  120,  the  product  of  the 

units   in   the  two  given  dimensions,  must  be  the 

number  of  feet  of  thickness  required.     480  -^  120  =  4.     Hence,  4  feet 

must  be  the  thickness  required. 

405.  Rules  for  Measurement  of  RecianguSar  Solids.— i.  Multi- 
ply the  length,  width  and  thickness  together,  and  the 
product  will  denote  the  cubic  contents. 

B.  Divide  the  cubic  contents  by  the  product  of  any 

two  of  the  dimensions,  and  the  qiootient  will  denote 

the  other  diinension. 
15  « 


174  REVIEW   PROBLEMS. 

probz,i:ms. 

1.  How  many  cubic  feet  is  the  capacity  of  a  bin  whose  in- 
side measures  12  feet  long,  6^  feet  wide  and  b\  feet  deep? 

Alls.  400. 

2.  What  are  the  contents  of  a  cube  wiiose  edge  measures 
5.5  feet?  Ans.  166.375  cu.  ft. 

3.  How  many  cord  feet  are  there  in  a  load  of  wood  8  feet 
long,  3|  feet  wide  and  5  feet  high  ? 

4.  How  much  wood  of  the  usual  length  is  there  in  a  range 
163  feet  long  and  4  feet  high  ? 

5.  If  a  load  of  wood  is  8  feet  long  and  3  feet  wide,  how 
high  must  it  be  to  contain  a  cord  ?  Ans.  5  ft.  4  in. 


SECTION  XLII. 

BEVIEW   PROBLEMS. 

mentaij  exercises. 

406. — Ex.  1.  How  much  will  2  pecks  of  berries  cost,  at  12^ 
cents  per  quart  ? 

2.  How  much  is  gained  by  selling  a  gross  of  buttons,  which 
cost  75  cents,  for  8^  cents  per  dozen  ? 

3.  If  the  forenoon  and  afternoon  sessions  of  a  school  are 
each  3  hours,  what  part  of  the  two  sessions  are  two  recesses  of 
20  minutes  each  ? 

4.  When  it  is  9  o'clock  in  the  morning  at  Philadelphia, 
what  time  is  it  at  a  point  15°  45'  west  of  Philadelphia  ? 

5.  How  much  must  be  paid  for  a  board  22  feet  long  and  1 
foot  6  inches  wide,  at  the  rate  of  $30  per  thousand  square  feet  ? 

6.  If  you  should  leave  home  and  travel  till  your  watch  is 
35  minutes  fast,  how  far  in  longitude  would  you  have  travelled, 
and  in  what  direction  ? 

7.  Two  boys  were  cni])loyed  to  measure  the  length  of  a  ditch  ; 
one  reported  it  to  be  1  rd.  16  ft.  11  in.,  and  tlic  other  1  rd.  5  yd. 
Ift.  11  in.  Tlic  true  length  was  2  rd.  5  in.  How  much  did 
cacii  of  the  mcasurenienls  differ  from  the  true  length? 


REVIEW   ITxOBLEMlS.  175 

WRITTEN    EXERCISES. 

407. — Ex.  1.  How  many  hogsheads  are  2217  quarts? 

2.  How  mauy  quarts  are  8  hhd.  50  gal.  1  qt.  ? 

3.  In  1000000  seconds  are  how  many  weeks  ? 

4.  In  1  wk.  4  d.  46  min.  40  sec.  are  how  many  seconds  ? 

5.  How  much  wood  in  a  range  40  feet  long,  1\  feet  high  and 
4  feet  wide  ?  Ans.  9f  cords. 

6.  What  must  be  the  height  of  a  range  of  wood  which  is 
40  feet  long  and  4  feet  wide,  to  contain  9  cd.  3  cd.  ft.'? 

7.  What  decimal  part  of  3  gallons  is  3  pints  ? 

8.  What  is  the  sum  of  f  of  a  foot,  f  of  a  yard  and  -J  of  a 
mile  ?  Am.  280  rd.  0  yd.  2  ft.  1\  in. 

9.  What  is  the  value  of  .131  of  5  hours? 

10.  What  decimal  part  of  5  hours  is  40  minutes? 

11.  How  many  years  after  the  battle  of  Lexington,  A})ril 
19,  1775,  was  that  of  New  Orleans,  January  8,  1815  ? 

12.  How  many  days  will  there  be  from  October  10,  1871,  to 
March  17,  1872? 

13.  If  the  16th  of  May  is  Sunday,  what  day  of  the  week  is 
the  20th  of  the  next  October  ? 

Solution. — The  difference  in  the  given  dates  is  157  days,  or  22  weeks 
3  days.  3  days  after  Sunday,  the  ending  of  the  22  weeks,  must  be 
Wednesday,  the  day  of  the  week  required. 

14.  What  part  of  a  day  is  6  h.  3  min.  4  sec.  ? 

15.  If  a  clock  tick  172800  times  a  day,  how  many  times 
will  it  tick  in  6  h.  3  min.  4  sec.  ? 

16.  When  2  bu.  1  j)k.  of  clover-seed  is  sold  for  $5.94,  what  is 
the  price  of  a  peck  ? 

17.  How  much  molasses  in  43  casks,  each  holding  97  gal. 
1  pt.  2  gi.  ?  Am.  4179  gal.  0  qt.  0  pt.  2  gi. 

18.  How  long  must  a  rectangular  lot  of  land  be,  whose 
width  is  16  rods,  to  contain  2  acres  ?  Ans.  20  rods. 

19.  How  much  is  f  of  45  T.  15  cwt.  25  lb.  ? 

20.  How  many  yards  of  cambric,  which  is  f  of  a  yard  wide, 
will  line  6f  yards  of  cloth,  which  is  1^  yards  wide  ? 

21.  What  will  it  cost  to  carpet  a  room  20  feet  wide  and  18 
feet  long  with  carpeting  4  feet  wide,  costing  $2.33|-  per  yard  ? 


176  PERCENTAGE. 

SECTION   XLIII. 
FERCEJ^'TAGE. 

408.— Ex.  1.  How  much  is  -^  of  100  yards?    yfj? 

2.  How  much  is  ^  of  100  ?     ^-^  ?     y-^^  ?    ilnr  ? 

3.  What  part  of  100  yards  is  1  yard?     Is  11  yards? 

4.  How  many  hundredths  of  a  hundred-weight  are  3  pounds? 

5.  How  many  hundredths  of  anything  is  \  of  it?     -^  of  it? 

6.  What  part  of  100  hundredths  is  25  hundredths? 

DEFINITIOETS. 

409.  Any  Per  cent,  of  a  number  is  so  many  hundredths  of 
that  number. 

The  term  j)e.r  cent,  is  a  contraction  of  the  Latin  per  centum,  and  means 
by  the  hundred. 

Thus,  6  per  cent,  of  25  is  .06  of  25  ;  and  5  per  cent,  of  4  is  .05  of  4. 

410.  The  Sign  %  is  generally  used  by  business-men,  instead 
of  the  words  per  cent. 

Thus,  8fo  means  8  per  cent. 

411.  Any  per  cent,  less  than  100  per  cent,  may  be  expressed 
by  a  decimal  or  common  fraction ;  any  per  cent,  equal  to  or 
greater  than  100  per  cent,  may  be  expressed  by  an  integer,  a 
mixed  number  or  an  improper  fraction  ;  and  a  fractional  part 
of  1  per  cent,  may  be  expressed  as  a  common  fraction  at  the 
right  of  the  figure  in  the  hundredths  order.     Thus, 

1  per  cent,  may  be  written,  ll 

6  per  cent.  " 

7  per  cent.  " 

7-j^  per  cent.  " 

12^  per  cent.  " 

100  per  cent.  " 

i^5  per  cent.  "  l^o'^.l.l^o.  "     —^ 


n,  n. 

.01, 

or 

100 

6%, 

.06, 

11 

6 

100 

7%, 

.07, 

i< 

7 
100 

n,3 

H^- 

07— 

•^'  10' 

« 

'To 
100 

1^1%, 

■i^i 

« 

^4 

100 

100%, 

1.00, 

" 

100 
inn 

PER  CENT  A  GE.  177 


WRITTEN  EXERCISES. 

412.  Write  and  read — 

1.  4%. 

2.  5%. 

3.  16%. 


2\  per  cent. 
16f  per  cent. 
217  per  cent. 


4.  ouyc 

5.  17%. 

6.  106% 

10.  Write  decimally  3%  ;  11%  ;  14%  ;  93%. 

11.  Write  as  a  common  fraction  5%  ;  17%  ;  31%. 

12.  Express  as  a  number  of  hundredths  \;  |- ;  f . 

13.  Express  in  per  cent.  | ;  | ;  2^ ;  6^ ;  4^. 

GENERAL  CASES  OF  PERCENTAGE. 

413. — Ex.  1.  To  take  5%  of  a  number  is  to  take  how  many 
hundredths  of  that  number  ? 

2.  What  is  5%  of  100  bushels?     Of  200  yards? 

3.  What  is  1%  of  $500?     7%  of  8500? 

4.  What  per  cent,  of    100   bushels  is  5  bushels?     Is  6 
bushels  ?     Is  25  bushels  ? 

5.  What  per  cent,  of  $500  is  $5  ?     $35  ? 

6.  Five  bushels  are  5  %  of  what  number  of  bushels  ? 

7.  Thirty-five  dollars  are  7  %  of  what  number  of  dollars  ? 

8.  Twelve  dollars  are  2%  of  Avhat  number  of  dollars? 

9.  Of  what  number  are  15  yards  6%  ? 

10.  Of  what  number  are  27  gallons  9%  ? 

11.  To  what  will  $400  amount  if  increased  by  5%  of  itself? 

12.  How  much  is  $200  diminished  by  25%  of  itself? 

13.  A  certain  number  increased  by  5%   of  itself  is  420; 
what  is  that  number  ? 

14.  A  certain  number  diminished  by  25%  of  itself  is  150 ; 
what  is  that  number  ? 

DEFINITIONS, 

414.  Percentage  is  the  process  of  computing  by  hundredths. 

415.  The  Base  is  the  number  or  quantity  of  which  the  hun- 
dredths are  computed. 

416.  The  Rate  Per  Cent.,  or  Rate,  is  the  number  denoting 
the  number  of  hundredths  of  the  base  which  are  taken. 


178  PERCENTAGE. 

417.  The  Percentage  is  the  result  of  finding  a  number  of 
hundredths  of  the  base. 

The  percentage  is  also  sometimes  called  the  per  cent. 

418.  The  Amount  is  the  base  with  the  percentage  added  to  it. 

419.  The  Difference  is  the  base  with  the  percentage  sub- 
tracted from  it. 

CA.SE   I. 

Base  and  Rate  giren,  to  find  the  Percentage,  Amount  or  Difference. 

420.— Ex.  1.  What  is  20%  of  455  yards,  and  what  is  the 
amount  and  difierence  ? 

Solution.  —  Since 

A55  vd  ^^^   ^^  ^"7  "i^^ber 

Qf^  is  .20  of  that  number, 

20%    of    455    yards 


Or, 


01.00yd.  j^^gj  be  .20  of  455 

yards,  or  91  yards. 

455  yd.  X  ^  =  y-  yd.  =^91  yd.  Or,  since  '20%    of 

any  number  is   i  of 

Amount  =  4^5  yd.  +  91  yd.  =  54-6.  that  number,  20%  of 

-r..r-  ,  -  <-     1        n-1      1         r> n  t     i       ^^^  yards  must  be  \ 

Dv^erence  =  400  yd.  -  91  yd.  =  064  yd.      ^^  455  y^^^,,^  ^^  g^ 

yards. 
The  amount  is  455  yards  plus  the  percentage,  or  546  yards ;  and  the 
difference  is  456  yards  minus  the  percentage,  or  364  yards. 

2.  How  much  is  15%  of  $460? 

3.  How  much  is  2%  of  6550  pounds,  and  what  is  the 
amount  ? 

421.  Principles. — 1.   The  percentage  is  equal  to  the  product 

of  the  base  by  the  rate. 

2.   The  amount  is  equal  to  the  sum  of  the  base  and  percentage. 
8.   The  difference  is  equal  to  the  base  less  the  percentage. 

422.  Rules  for  Finding  the  Percentage,  Amount  or  Difference.— 

1.  Multiply  the  base  by  the  rate,  and  the  produet  will 
be  the  percentage. 

2.  Add  the  percentage  to  the  base,  and  the  sum  will 
be  the  amount;  or  subtract  the  percentage  from  the 
base,  and  the  result  will  be  the  difference. 


PERCENTAGE. 


179 


I'll  OB  LEMS. 


Find  the  percentage  of — 

1.  516  bushels  at  5%. 

2.  $360  at  15%.        Ans.  $54. 

3.  455  gallons  at  3%. 

4.  93  acres  at  7%.  Ans.  6.51  A. 

5.  812  men  at  25%. 


How  much  is — 

6.  7%  of  210  pounds? 

7.  1%  of  703  yards? 

8.  33^%  of  942  sheep? 

9.  125%  of  1215  tons? 
10.  350%  of  $1600? 


11.  If  a  man,  whose  income  is  $1250  a  yeai",  spends  80% 
of  it  for  his  living,  how  many  dollars  does  he  spend  ? 

Ans.  $1000. 

C^SE   II. 
Base  and  Percentage  given,  to  find  the  Rate. 

423.— Ex.  1.  What  per  cent,  of  455  yd.  is  91  yd.? 


91  _ 

455' 
Or, 

91 


.20  =  20%  Solution.— 91  yd.  are  /^V  of  455  yd.,  or 

.20,  or  20%  of  455  yd.     Or,  91  yd.  are  ^V^ 
of  455  yd.     -^-^-^  =  .20 ;  hence,  91  yd.  equal 


1^.  of  100%  =20%     -20  of  455  yd.,  or  20%. 

2.  What  per  cent,  of  $390  is  $11.70?  Ans.  3. 

3.  What  per  cent,  of  3240  T.  is  21.6  T.  ?         Ans.  |  of  1. 

42-i:.  Principle. —  The  rate  is  equal  to  the  quotient  of  the  per- 
centage divided  by  the  base. 

425.  Rules  for  Finding  the  Rate  Per  Cent.— i.  Divide  the  per- 
centage by  the  base.    Or, 

2.  Tahe  such  a  part  of  100  per  cent,  as  the  percent- 
age is  of  the  base. 


PMOBI^JEMS. 


What  per  cent,  of — 

1.  516  bu.  is  25.80  bu.  ? 

2.  455  gal.  is  13.65  gal.? 

3.  93  A.  is  6.51  A.  ? 

4.  1.4.701b.  is  14.701b.? 


5.  $1600  is  $4000?  Ans.  250.      10.  $6300  is  $173.25? 


6.  460ft.  is  368 ft.?  ^ns.  80. 

7.  51 4 i  rd.  is  84  rd.  ? 

8.  942  sheep  is  314  sheep  ? 

9.  703yd.  is  5.261  yd.? 


180  PERCENTAGE. 

11.  What  per  cent,  of  2  A.  72  sq.  rd.  is  144  sq.  rd.? 
Solution. 
2  A.  72  sq.  rd.  =  S92  sq.  rd. 

12.  A  pound  Troy  is  what  per  cent,  of  a  pound  Avoirdupois  ? 

13.  A  merchant  had  1  T.  15  lb.  10  oz.  of  sugar,  and  sold 
10  cwt.  46  lb.  4  oz.    What  per  cent,  did  he  sell  ?    Ans.  51|f . 

CASE    III. 

Rate  and  Percentag-e  Given,  to  find  tlie  Base. 

426.— Ex.  1.  91  yd.  are  20%  of  how  many  yards? 

_9i  _    .  _.  _  Solution. — 20%  of  any  number  is  .20  of  that 

'  'SO      "^  number.     Since  91  yd.  are  20  %  of  some  number, 

Or  1  %   of  tliat  number  must  be  i^^  of  91  yd. ;  and 

Q-]  \/  FT  -^  lepr  100%,  or  the  number  required,  must  be  100 
times  -^-^  of  91  yd.,  or  V'j*'  of  91  yd.,  which  is  the 
same  as  the  quotient  obtained  by  dividing  91  yd.  by  .20.     Or, 

Since  100%  is  5  times  20%,  and  91  yd.  are  20%  of  some  number,  that 
number  must  be  5  times  91  yd. ;  or,  since  20%  is  |  of  a  number,  and  91  yd. 
are  20%  of  some  number,  that  number  must  be  the  quotient  obtained  by 
dividing  91  yd.  by  \. 

2.  Of  what  number  are  71  mi.  16|%  ? 

3.  Of  what  number  are  203  men  50%  ?       Ans.  406  men. 

427.  Principle. —  The  hose  is  equal  to  the  quotient  of  the  per- 
centage divided  by  the  rate. 

428.  Rules  for  Finding  the  Base  of  the  Percentage.— i.  Divide 
the  percentage  hy  the  rate.    Or, 

2.  Tahe  as  Tnany  times  the  percentage  as  100  per 
cent,  is  times  the  rate. 

PJlOIiLJSMS. 

Of  what  are — 

1.  13.65  gal.,  3%  ?  4.  5.27J  yd.,  f  %  ? 

2.  6.51  A.,  7%  ?  5.  85600,  350%  ? 

3.  822.61, 1%  ?    A71S.  $2584.  6.  1724  ft.,  400%?  Ans.  431  ft. 


PERCENTAGE.  181 

7.  If  the  percentage  be  23.8  lb.  and  the  rate  4%,  what  must 
be  the  base  ? 

8.  If  the  rent  of  a  store  at  $1020  is  12%  of  its  valuation, 
what  is  its  valuation  ?  Ans.  $8500. 

9.  A  farmer  sold  85  A.  100  sq.  rd.  of  land,  which  was  just 
25%  of  his  farm.     What  was  the  extent  of  his  farm  ? 

C^SE    TV. 

Amount  or  Difference  and  Rate  Given,  to  Find  the  Base. 

429. — Ex.  1.  A  certain  number  increased  by  8%  of  itself  is 
189  ;  what  is  that  number? 

1.08)189.00(175 

108 

- Solution. — A  number  increased  by  8%  of  it- 

810  self  is  108%  of  itself,  or  1.08  times  itself. 

756"  If  189  is  1.08  times  a  number,  that  number 

r  irx  must  be  y.^-j  of  189,  or  175. 

540 

2.  A  certain  number  diminished  by  25%  of  itself  is  327. 
What  is  the  number  ? 

.75)327.00(4.36 
300 

Solution. — A  number  diminished  by  25%  of 

270  itself  is  75%  of  itself,  or  .75  of  itself. 

225  If  327  is  .75  of  a  number,  that  number  must  be 

-J^  .^\  of  327,  or  436. 

.  450 

3.  If  the  amount  is  124.20,  and  the  rate  15%,  what  is  the 
base?  Am.  108. 

4.  If  the  difference  is  278.30,  and  the  rate  45%,  what  is  the 
base? 

430.  Principle. —  The  base  is  equal  to  the  quotient  of  the 
amount  divided  by  1  plus  the  rate,  or  to  the  quotient  of  the  differ- 
ence divided  by  1  minus  the  rate. 

431.  livi\e.— Divide  the  amount  by  1  plus  the  rate,  or 
divide  the  difference  by  1  minus  the  rate. 

16 


182  PERCENTAGE. 

PROBLEMS. 

1.  An  infantry  regiment,  after  losing  1\%  of  its  men,  had 
740  left.     How  many  had  it  at  first  ?  Ans.  800. 

2.  By  running  15%  faster  than  usual,  a  locomotive  ran  552 
miles  in  a  day.     What  was  the  usual  daily  speed  ? 

Ans.  480  miles. 

3.  A  farmer  purchased  a  farm  for  a  certain  sum,  expended 
for  tools  and  stock,  11%  of  the  price  of  the  farm,  and  found 
that  the  whole  cost  was  $7215.  What  was  the  cost  of  the 
farm  alone? 

4.  Osgood  raised  800  bushels  of  corn,  which  was  25%  more 
than  ^  of  what  Benton  raised.  How  many  bushels  did  Ben- 
ton raise?  Ans.  1280. 


SECTION   XLIV. 

PROFIT  AKB  LOSS. 

432. — Ex.  1.  I  sold  a  barrel  of  flour,  which  cost  $8,  at  an 
advance  of  25  % ,  or  for  \  more  than  the  cost.  How  much  did 
I  gain? 

2.  John  bought  a  hat  for  $5,  and  sold  it  at  a  loss  oi1^%,  or 
for  "I"  less  than  the  cost.     How  much  did  he  lose  ? 

3.  If  I  sell  a  horse,  which  cost  me  $120,  at  a  profit  of  10%, 
how  much  do  I  get  for  him  ? 

4.  I  had  40  sheep,  but  have  sold  4  of  them.  What  per 
cent,  of  the  40  have  I  sold  ? 

5.  What  per  cent,  shall.  I  gain  by  selling  for  §10,  flour 
which  cost  me  $8  ? 

6.  What  per  cent,  does  a  merchant  lose  by  selling  goods  at 
■|  of  their  cost  ? 

7.  I  sold  flour  at  $10  and  gained  25  % .    What  did  it  cost  me  ? 

DEFINITIONS. 

433.  Profit  and  Loss  are  terms  used  to  denote  the  gain  or 
loss  in  businci^s  transactions. 

The  gain  or  loss  may  be  regarded  as  a  certain  per  cent,  of 
the  cost.     Hence, 


PBOFIT  AND  LOSS.  183 

434:.  The  Base  of  computation  of  profit  or  loss  is  the  cost ; 
and,  by  the  principles  of  percentage,  we  have  the — 

435.  Rules  for  Profit  or  Loss.— _Z.  Multiply  the  cost  by  the 
rate,  and  the  product  will  he  the  profit  or  loss. 

2.  Divide  the  profit  or  loss  by  the  cost,  and  the  quo- 
tient will  be  the  rate. 

3.  Multiply  the  cost  by  1  plus  the  rate  of  profit,  or  by 
1  minus  the  rate  of  loss,  and  the  product  will  be  the 
selling  price. 

Jj..  Divide  the  profit  or  loss  by  the  rate;  or,  divide  the 
selling  price  by  1  plus  the  rate  of  profit,  or  by  1  minus 
the  rate  of  loss,  and  the  quotient  ivill  be  the  cost. 

I'JiOBLJEMS. 

Ex.  1.  I  bought  a  house  for  83500,  and  sold  it  at  a  gain  of 
12|-%.     How  much  was  the  profit? 

2.  Goods  which  cost  85400,  were  sold  at  9%  below  cost. 
How  much  was  the  loss?  Ans.  $486. 

3.  A  farmer  had  460  sheep,  which  cost  him  $3  each,  but  he 
lost  5%  of  them.     How  much  was  the  loss?  Ans.  $69. 

4.  A  merchant  bought  112  barrels  of  flour,  at  87  a  barrel, 
and  sold  it  so  as  to  gain  15%.     How  much  was  the  profit? 

5.  How  much  will  be  my  loss  on  3  casks  of  molasses,  of  63 
gallons  each,  which  cost  me  80  cents  a  gallon,  if  I  am  obliged 
to  sell  it  at  10%  below  cost? 

6.  Williams  bought  coal  at  $7.50  per  ton,  and  sold  it  at 
$7.05  per  ton.     What  was  the  loss  per  cent.  ?       .        Ans.  6. 

7.  A  field  produced  the  value  of  867  one  year,  and  816.75 
more  the  next  year.     What  was  the  gain  per  cent.  ? 

8.  If  I  sell  1^  of  a  house  at  f  of  the  cost  of  the  whole  house, 
do  I  gain  or  lose,  and  at  what  rate?  Ans.  Gain  20%. 

9.  By  selling  coal  at  88.05  per  ton,  I  gained  15%.  What 
v/as  the  cost  per  ton  ?  Atis.  87. 

10.  A  watch  not  proving  as  good  as  I  expected,  I  was  con- 
tent to  sell  it  at  a  loss  of  $6,  which  was  7^%  of  the  cost. 
What  was  the  cost?  Ans.  $80. 


184  PROFIT  AND  LOSS. 

11.  By  selling  an  article  at  87|^%  of  its  cost,  I  lost  $125. 
What  was  the  cost  ? 

12.  I  sold  molasses  at  115%  of  its  cost,  and  thereby  gained 
9  cents  on  a  gallon.     What  was  the  cost  per  gallon  ?    ' 

13.  Henry  bought  cloth  at  $6.50  per  5^ard ;  at  what  price 
must  it  be  marked  to  allow  of  12%  profit  ? 

14.  At  what  price  must  I  sell  a  house  which  cost  me  $5400, 
to  gain  20%  ?  Ans.  $6480. 

15.  John  paid  $225  for  a  horse,  and  $15  for  his  keeping. 
For  how  much  must  he  be  sold  above  the  first  cost  to  allow  of 
5|%  profit  ?  Ans.  $28.20. 

16.  I  bought  60  yd.  of  cloth  for  $240  ;  Avhat  must  be  my 
selling  price  per  yard  to  make  12^%  ? 

17.  I  sold  a  house  for  $6976,  and  thereby  gained  9%.  What 
was  the  cost  ?  Ans.  $6400. 

18.  By  selling  tea  at  76  cts.  per  pound,  I  sufier  a  loss  of  20%. 
What  was  the  cost  ? 


TEST    QUESTIONS. 

436. — 1.  What  is  any  Per  Cent,  of  a  number?  Of  what  is  the  term 
a  contraction  ?    What  sign  is  generally  used  for  the  words  Per  Cent.  ? 

2.  How  may  any  per  cent,  less  than  100  per  cent,  be  expressed  ?  A 
per  cent,  equal  to  or  greater  than  100  per  cent.  ? 

3.  What  is  Percentage?  The  base  of  percentage ?  The  rate?  The 
amount?     The  difference? 

4.  To  what  is  the  percentage  equal  ?  How  is  the  percentage  found 
when  the  base  and  rate  are  given  ?  How  is  the  amount  found  ?  How  is 
the  difference  found  ? 

5.  To  what  is  the  Rate  equal  ?  How  is  the  rate  found  when  tlie  base 
and  percentage  are  given? 

6.  To  what  is  tlie  Base  equal?  How  is  the  base  found  when  the  rate 
and  percentage  are  given  ?  Plow  is  the  base  found  wlien  the  amount  or 
difference  and  rate  are  given  ? 

7.  Wliat  are  Profit  and  liOss?  What  is  the  base  of  computation  of 
profit  and  loss?  To  what  is  the  profit  or  loss  equal ?  The  rate?  The 
cost  ? 

8.  To  what  is  tiie  Sei.ling  Price  equal?  How  is  the  cost  found 
when  the  selling  price  and  the  rule  uf  pruiit  ur  loss  are  known  ? 


COMMISSION. 


185 


SECTION     XLV. 

COMMISSIOJf. 

437.— Ex.  1.  How  much  should  I  receive  for  selling  goods 
to  the  amount  of  ^500,  if  allowed  2^  %  ? 

2.  If  I  am  allowed  2|%  for  making  purchases,  how  much 
per  ton  should  I  receive  for  purchasing  coal  at  $8  per  ton  ? 

3.  To  how  much  will  a  collector  be  entitled  for  collecting 
500,  at  2%  ? 

4.  How  much  will  remain  of  a  collection  of  $300,  after  de- 
ducting the  collector's  fees  at  the  rate  of  1^  %  ? 

DEFINITIONS. 

438.  An  A^cnt,  Comuiission  Merchant  or  Broker  is  a  person 
who,  by  authority,  buys  or  sells  goods  or  property,  or  collects 
money  for  another, 

439.  A  Consignee  is  a  person  to  whom  goods  are  sent  for 
sale,  and  a  Consignor  is  the  person  sending  the  goods. 

440.  Commission  is  the  percentage  allowed  an  agent  or  com- 
mission merchant  as  pay  for  transacting  business. 

441.  The  Base  of  commission  is  the  sum  expended  or 
received. 

442.  The  Amount  is  the  sum  expended  or  received  plus  the 
commission. 

443.  The  Net  Proceeds  of  a  sale  or  collection  are  the  sum  left 
after  the  commission  and  other  charges,  if  any,  are  deducted. 

444.  Rules  for  Commission.— i.  Multiply  the  base  by  the 
rate,  and  the  product  is  the  coTmnission. 

2.  Subtract  from  the  base  the  cojmnission,  and  the 
other  charges,  if  any,  and  the  result  will  be  the 
proceeds. 

3.  Divide  the  coinmission  by  the  base,  and  the  quo- 
tient is  the  rate. 

Jf.  Divide  the  commission  by  the  rate,  or  divide  the 
aTYiount  by  1  plus  the  rate,  aivd  the  quotient  is  the 
hojse. 

16* 


186  COMMISSION. 

PROnLEMS. 

1.  An  agent  has  sold  goods  for  me  to  the  amount  of  $5000, 
at  2^%  commission.    What  is  his  commission  ?     Atis.  $125. 

2.  A  commission  merchant  sells  4520  bu.  of  wheat,  at  $2.50 
per  bushel.     How  much  is  his  commission  at  3%  ? 

3.  If  I  collect  as  an  agent  $390,  and  am  entitled  to  5% 
commission,  how  much  must  I  pay  over  ?  Ans.  $370.50. 

4.  An  agent  bought  80  barrels  of  beef,  at  $22  per  barrel, 
and  paid  $16  for  insurance  and  $9  for  cartage.  His  commis- 
sion v\-as  2^%.  What  was  the  amount  of  his  bill  to  his 
employer?  Ans.  $1824.60. 

5.  A  commission  merchant  having  sold  some  goods,  paid 
$4168.80  to  his  employer,  and  retained  as  his  commission 
$151.20.      What  was  the  rate  of  his  commission  ? 

6.  My  agent  collected  $390,  and,  retaining  his  commission, 
paid  over  $370.50.     At  what  rate  was  his  commission  ? 

7.  A  commission  merchant  bought  some  goods,  paid  for 
cartage  $21.50,  and  charged  for  storage  $31,  and  for  commis- 
sion $112.50.  His  entire  bill  was  $5165.  What  was  the  rate 
of  commission?  Ans.  2^%. 

8.  I  paid  a  commission  merchant  $12.56  for  selling  goods, 
at  the  rate  of  4%.     What  amount  did  he  sell?    Ans.  $314. 

9.  A  treasurer's  commission  for  collecting  taxes  in  one  year, 
at  1|%,  is  $413.10.     What  was  the  amount  collected? 

10.  A  commission  merchant  in  Chicago  received  $1665.62|- 
with  which  to  purchase  flour  at  $6.50  per  barrel,  after  deduct- 
ing his  commission  of  2|^%.  How  much  was  his  commission, 
and  how  many  barrels  did  he  purchase? 

Ans.  Commission,  $40,621;  barrels,  250. 

11.  An  agent  received  $5922  with  which  to  purchase  goods, 
after  deducting  his  commission  of  5%.  How  much  was  his 
commission,  and  what  was  the  sum  to  be  expended? 

Ans.  Commission,  $282  ;  sum,  $5640. 

12.  I  remit  to  an  agent  $360.70,  with  which  to  purchase 
goods;  deduct  his  commission  of  5%,  and  pay  $3.70  for 
insurance.     What  sum  can  he  expend  for  the  goods  ? 


INSURANCE.  187 

SECTION    XLVI. 
IJfSUBAJ^CE. 

445.  Insurance  is  indemnity  secured  for  loss. 
446.-  Fire  IiisiuMnce  is  indemnity  secured  for  loss  of  property 
by  fire  or  lightning. 

447.  Marine  Insurance  is  indemnity  secured  for  loss  of  prop- 
erty by  casualties  of  navigation. 

448.  Health,  or  Accident,  Insurance  is  indemnity  secured  for 
loss  by  sickness  or  accident. 

449.  Life  Insurance  is  indemnity  secured  for  loss  by  death. 

450.  The  Policy  is  the  contract  between  the  insurer  and  the 
insured. 

451.  The  Premium  is  the  sum  paid  for  insurance. 

452.  In  Property  Insurance  the  premium  is  computed  at  a 
certain  rate  per  cent,  on  the  value  insured. 

453.  In  Life  Insurance  the  premium  is  computed  at  a  certain 
sum  or  rate  per  |100  or  $1000  insured. 

454.  Rules  for  Insurance.— i.  Multiply  the  value  insured 
by  the  rate,  and  the  product  is  the  premium. 

2.  Divide  the  premium  by  the  value  insured,  and, 
the  quotient  is  the  rate. 

3.  Divide  the  value  insured  by  1  minus  the  rate, 
and  the  quotient  luill  be  the  ainount  to  be  insured  to 
cover  the  value  insuj^ed,  and  pj-emimn  of  insurance. 

PliOBT.EMS. 

1.  What  premium  must  be  paid  for  insurance  of  $3000  on 
a  house,  at  2^%  ?  Ans.  $75. 

2.  What  is  the  expense  of  insuring  f  of  a  mill  valued  at 
$8400,  at  5^^,  the  policy  being  $1  ?  Ans.  $316. 

8,  Hall  paid  $91,  including  the  policy  at  $1,  for  insuring 


188  INSURANCE. 

$2500  on  his  house  and  $2000  on  his  saw-mill.     What  was  the 
rate  of  insurance  ?  Ans.  2  % . 

4.  If  $75  is  paid  for  insuring  $3000  on  a  house,  what  is  the 
rate  of  premium  ? 

5.  What  sum  must  be  insured  on  $5600,  at  3%,  to  cover 
property  and  premium  in  case  of  loss?       Ans.  $5773.19  +. 

6.  For  what  sum  must  property,  valued  at  $4000,  be  insured, 
at  5%,  to  cover  f  of  the  property,  the  premium  and  the  policy 
at  $2? 

7.  A  man  40  years  old  has  obtained  a  life  policy  for  $6000, 
at  the  rate  of  $29.60  per  $1000.    What  is  the  annual  premium  ? 

8.  A  man  31  years  old  took  out  a  life  policy  for  $7500  for 
the  benefit  of  his  family,  on  the  plan  of  semi-annual  pay- 
ments, at  the  rate  of  $23.78  per  $1000.  He  died  at  the  age 
of  35.  How  much  did  the  amount  due  his  family  exceed  the 
payments  he  had  made  ?  Ans.  $6073.20. 

9.  If  $3160  must  be  insured  on  a  house  to  cover  |  of  its 
value,  the  premium  at  5%  and  the  policy  at  $2,  what  is  the 
value  of  the  house  ? 


TEST  QUESTIONS. 

455.— 1.  What  is  a  Cojimission  Merchant  or  Broker  ?    A  con- 
signee?    A  consignor? 

2.  What  is  Commission  ?     The  base  of  commission  ?    The  amount  ? 
What  are  the  net  proceeds  ? 

3.  To  what  is  the  Commission  equal  ?     How  is  it  found  when  the  base 
and  rate  are  given  ? 

4.  To  what  are  the  Net  Proceeds  equal  ?  How  are  they  found  when 
the  base  and  commission  are  given  ? 

5.  To  what  is  the  Kate  equal  ?    How  is  the  rate  found  when  the  com- 
mission and  base  are  given  ? 

6.  To  what  is  the  Base  equal  ?    How  is  it  found  when  the  commission 
■and  rate  are  given  ?     When  the  amount  and  rate  are  given  ? 

7.  What  is  Insurance?    Fire  insurance?   Marine  insurance?   Health 
and  accident  insurance  ?     Life  insurance  ? 

8.  What  is  the  Policy  ?     The  premium  ?    How  is  the  premium  com- 
puted in  property  insurance  ?    In  life  insurance  ? 


BEVIi:W   PROBLEMS.  189 

SECTION    XLVII. 
REVIEW  PROBLEMS. 

MJENTAZ    EXEKCISES. 

456. — Ex.  1.  At  what  price  must  tea  which  cost  75  cts.  be 
sold,  to  make  16f%  profit? 

2.  I  have  60  cts. ;  how  much  of  it  must  I  spend  to  have  88% 
of  it  left? 

3.  If  I  had  35  sheep  and  sold  14,  what  per  cent,  of  them 
did  I  sell  ? 

4.  I  bought  a  cart  for  $75,  and  sold  it  for  $90.  What  per 
cent,  did  I  gain  ? 

6.  In  a  certain  school,  5  of  every  8  pupils  are  girls.  What 
per  cent,  are  boys  ? 

6.  69^  of  a  number  is  what  per  cent,  of  30%  of  the  number? 

7.  A  merchant  sold  tea  at  a  loss  of  IG  cts.  a  pound,  which 
was  25%  of  the  cost.    What  was  the  cost  ? 

8.  Henry  is  11  yr.  old  and  John  is  125%  as  old.  How 
many  years  is  the  difference  in  their  ages  ? 

9.  If  37|-%  is  lost  by  selling  goods  for  $90,  what  was  their 
cost? 

10.  75  is  25%  more  than  what  number?  42  is  20%  more 
than  what  number  ? 

11.  What  amount  of  money  must  be  forwarded  to  a  com- 
mission merchant,  to  cover  a  purchase  of  $180  and  his  com- 
mission of  10%? 

12.  A  horse  was  sold  for  $90,  at  which  price  12  J  %  was 
gained.  What  per  cent,  would  have  been  gained  by  selling 
him  for  $100  ? 

13.  I  bought  a  wagon  for  $72,  which  Avas  20%  more  than 
its  value.  I  sold  it  at  5%  less  than  its  value.  How  much  did 
Hose? 

14.  Higgins  sold  a  cow  for  $30,  and  by  the  transaction  lost 
16-|%.  He  sold  another  cow  at  an  advance  of  16%  for  just 
enough  to  cover,  by  the  profits,  the  loss  upon  the  first  cow. 
What  did  he  get  for  the  last  cow  ? 


190  REVIEW    PROBLEMS: 

WniTTEN  EXERCISES. 

457^ — Ex.  1.  If  a  cubic  foot  of  pine  timber  weighs,  when 
green,  44  lb.  12  oz.,  and  when  seasoned,  30  lb.  11  oz.,  what  per 
cent,  does  it  lose  in  seasoning  ?  A^is.  31y^^. 

2.  How  large  a  sale  must  a  merchant  make,  at  a  profit  of 
15%,  to  clear  $3750?  Ans.  $25000. 

3.  What  must  be  the  selling  price  and  profit  of  coal  whose 
first  cost  is  $6,  freight  10%  and  rate  of  gain  20%  ? 

A}is.  Selling  price,  $7.92 ;  profit,  $1.32. 

4.  If  a  merchant  closed  out  his  goods  at  a  loss  of  10%,  how 
much  did  he  lose  on  calico  that  cost  12^  cts.  per  yard,  and  on 
sugar  that  cost  15  cts.  per  pound? 

Ans.  On  calico,  1^  cts.  per  yard. 
On  sugar,  1|  cts.  per  pound. 

5.  My  horse  cost  |  as  much  as  my  carriage ;  what  per  cent, 
of  the  cost  of  the  one  was  the  cost  of  the  other  ? 

Ans.  The  horse,  60%  of  the  carriage. 
The  carriage,  166|%>  of  the  horse. 

6.  I  sold  one  half  of  a  lot  of  goods  which  cost  me  $456,  at  a 
loss  of  25%,  and  the  other  half  at  a  profit  of  $69.54.  What 
was  the  gain  per  -cent,  on  the  whole  transaction?      Am.  2|. 

7.  I  bought  a  farm  for  a  certain  sum,  and  after  expending 
5|%  of  the  cost  for  repairs  and  improvements,  and  paying  a 
tax  of  li%,  I  sold  it  for  $6420,  which  just  made  up  what  I 
had  paid  out.    What  was  the  original  cost  ?        ^ns.  $6000. 

8.  By  selling  cloth  at  $6  per  yard,  I  gain  25%?.     What  per 

cent,  shall  I  gain  by  selling  it  at  $5.28  ? 

Solution. — If  the 

gain  at  $6  per  yard 

1.25)$6.0000($4-80  is  25%,  $6  must  be 

goo  125%,  or  1.25  times 


1000 
1000 


the  cost,  and  the  cost 
must  be  yVu  of  $6, 

or  $4.80. 

By     selliiif;;    that 

.28-$4.80^i.48;   -§^  =  .10^10%    which  cost  $4.80  at 

^^^  $5.28,  the  gain  is  $.48, 

which  is  10%  of  $4.80, 


RE  VIE  W  PR  OBLEMS. 


191 


9.  A  man  drew  from  a  bank  $264,  which  was  8^%  of  his 
deposit.     What  was  his  deposit  ? 

10.  I  sold  a  watch  for  $69,  and  lost  20  per  cent.  What  per 
cent,  should  I  have  gained  if  I  had  sold  it  for  $93.50  ? 

11.  Do  I  make  or  lose  in  selling  an  article  marked  25% 
above  cost,  if  I  deduct  20%  ? 

Soi>TTTiON.— The  marked  price  is  25%  above  cost,  or  125%  of  cost 
20%,  or  \  of  125%  of  cost,  is  25%  of  cost.  125%  —25%  of  cost  is 
100%  of  cost,  or  the  cost.     Hence,  I  neither  make  nor  lose. 

12.  I  sold  goods  marked  40%  above  cost,  at  a  deduction  of 
35%.     What  per  cent,  did  I  lose?  Ans.  9. 

13.  A  merchant  sold  two  bills  of  goods  of  $75  each ;  on  the 
one  he  made  20%,  and  on  the  other  he  lost  20%.  What  was 
his  gain  or  loss  ?  Ans.  Loss,  $6.25. 

14.  Find  the  net  proceeds  of  the  following  account  of  sales, 
rendered  by  Barnard  &  Smith,  commission  merchants,  to 
Henry  Law,  consignor : 

Sales  of  Wheat  for  Acct.  of  Henry  Law,  Elgm, 


1871. 

To  WHOM  Sold. 

Description. 

Price. 

Jan. 

4 

Albert   Ward. 

101  hi. 

No.  1  Spring. 

$1.50 

$151 

50 

ti 

8 

Snyder  &  Co. 

552   " 

Mixed  Spring. 

1.40 

772 

80 

Feb. 

3 

0.  Sinith  &  Co. 

810    " 

Amber  State. 

1.45 

449 

50 

" 

10 

H.  A.  Stein. 

500    " 

White. 

1.60 

800 

00 

" 

17 

Thomas  Prince. 

75    " 

Mixed  Spring. 

1.40 

105 

00. 

Mar. 

2 

H.  Dunster. 

124    " 

Amber  State, 

1.45 

179 

80 

CHAEGES. 

$2458 

60 

Freight  and  Drc 
Insurance  on  $2( 

$51.00 
80.00 
11.00 

122.98 

214 

100  @  lj% 

Commission,  5% 

,  on  $2458.60 

Net  proceeds 

98 

Barnard  &  Smith. 


Chicago,  March  4, 1871. 


192  SIMPLE  INTEREST. 

SECTION    XLVIII. 
SIMPLE  lA^TEREST. 

458. — Ex.'l.  When  the  allowance  for  the  use  of  money  is  6 
per  cent.,  how  many  hundredths  of  the  money  is  the  allowance? 

2.  When  the  allowance  for  the  use  of  money  is  7  per  cent., 
how  many  hundredths  of  the  money  is  the  allowance  ? 

3.  When  the  allowance  for  the  use  of  money  is  5  per  cent, 
per  year,  what  is  the  allowance  for  the  use  of  $1  for  1  year  ? 
For  2  years  ?     For  3^  years  ? 

4.  When  the  allowance  for  the  use  of  money  is  6  per  cent, 
per  year,  how  many  hundredths  of  the  money  is  it  for  12 
months  ?     For  2  months  ? 

5.  At  the  rate  of  6  per  cent,  for  the  use  of  money  per  year, 
to  how  much  will  $300  amount  in  2  years  ? 

6.  When  the  allowance  for  the  use  of  $200  per  year  is  $30, 
W'hat  is  the  yearly  rate  ? 

7.  If  I  should  lend  $500  at  a  yearly  rate  of  4  per  cent,  for 
its  use,  to  how  much  would  it  amount  in  2\  years  ? 

DEFINITIONS. 

459.  Interest  is  an  allowance  for  the  use  of  money. 

460.  The  Principal  is  the  sum  for  the  use  of  which  interest 
is  paid. 

461.  The  Amount  is  the  sum  of  the  principal  and  interest. 

462.  The  Rate  of  Interest  is  the  rate  per  cent,  of  the  prin- 
cipal allowed  for  its  use  one  year. 

463.  A  Legal  Rate  of  Interest  is  any  rate  allowed  by  law, 
and  Usury  is  interest  reckoned  at  a  higher  rate  than  the  law 
allows. 

When  in  a  contract  between  parties,  no  rate  of  interest  is  named,  in 
most  of  the  States  and  on  debts  due  the  United  States,  the  legal  rate  is 
6%  ;  in  some  of  tlio  States  it  is  7^c,  in  several  ^'/c,  and  in  others  10%. 

In  this  book,  when  no  particular  rate  is  named  or  implied,  Qfo  is 
understood. 


SIMPLE  INTEREST.  193 

464.  In  the  Computation  of  interest  it  is  customary  to  reckon 
30  days  a  month,  and  12  months,  or  360  days,  a  year ;  and  in 
finding  the  difterence  between  dates,  to 

Take  the  number  of  entire  calendar  ino)iths,  and  the  actual 
number  of  days  left. 

Thus,  from  January  20  to  August  4  is  6  mo.  15  da.,  or  6^  mo. 

Months  are  often  conveniently  expressed  as  twelfths  of  a 
year,  and  3  days  as  a  tenth  of  a  month. 

Thus,  6  mo.  15  da.  may  be  expressed  as  G.5  mo.,  or  as  j^  of  a  year. 

465.  In  the  process  of  reckoning  interest,  partial  results,  if 
necessary,  may  be  carried  to  four  orders  of  decimals.  But  in 
answers  it  is  sufficiently  exact  to  reject  mills  if  less  than  5, 
and  if  5  or  more  than  5,  to  call  them  1  cent, 

466.  Simple  lutei'est  is  interest  on  the  principal  alone. 

CASE  I. 

Principal,  Rate  and  Time  Given,  to  Find  the  Interest  or  Amount. 

GENERAL  METHOD. 

467. — Ex.  1.  What  is  the  interest  and  what  is  the  amount 
of  $576  for  2  y.  6  mo.,  at  6%  ? 

$576 
.06 

^3 J-  56  Solution. — 2  y.  6  mo.  are  2|  years. 

■    q1  The  interest  of  $576  for  1  year  at  6%  is  .00  of  $576, 

2  or  $34.56. 

$69.12  Since  the  interest  for  1  year  is  $34.56,  for  2^  years  it 

1 7  28       "^"•''*  ^®  ^^  ^^^^^  $34.56,  or  $86.40. 

'- The  principal  added  to  the  interest,  or  the  amount,  ia 

$86.40       $662.40. 

576.00 


362.40 


2.  What  is  the  interest  of  $760  for  4  y.  8  mo.  ? 

3.  What  is  the  interest  of  $662.50  for  3 y.  4  mo.? 

17 


194  SIMPLE  INTEREST. 

4.  What  is  the  interest  of  §480.50  for  2  y.  7  mo.  12  da.,  at 
8%  ? 

$480.50 
.08 

$38  AAOO  Solution. — 2y.  7  mo.  12  da.  are  31.4  mo., 


31.4 


y. 

The  interest  of  $480.50  for  1  year,  at  8^,  is 
15376  .08  of  $480.50,  or  $38.44. 

3844  Since  the  interest  for  1  year  is  $38.44,  for 

i  1  notp  31.4  mo.,  or  ^\--^  of  a  year,  it  must  be  ^2*  of 

,  ^ — $38.44,  or  $100.58. 

12)$1207.016 

$100.58 

5.  What  is  the  interest  (Tf  $78.50  for  3  y.  3  mo.,  at  7%  ? 

6.  What  is  the  amount  of  $110.25  for  1  y.  8  mo.,  at  6^^  ? 

Ans.  $121.28. 

468.  Principle. — The  interest  is  equal  to  the  product  of  the 
principal,  rate  and  time. 

469.  Rules  for  Interest  by  the  General  Method.— i.  Multiply 
the  principal  hy  the  rate, -and  that  product  by  the  time 
expressed  as  years.    Or, 

2.  Multiply  the  principal  hy  the  rate,  and  that 
product  by  the  time  expressed  as  months,  and  divide 
the  result  by  12. 

S.  Add  the  interest  and  principal,  and  the  result 
will  he  the  amount. 

mOIiLJEMS. 

What  is  the  interest  of — 

1.  85631  for  1  year,  at  6%  ?  Ans.  $337.86. 

2.  $860  for  2  years,  at  7%  ?  Am.  $120.40, 

3.  $325for  5years,  at8%? 

4.  $1450  for  2  years  6  months,  at  6%  ?  Ans.  $217.50. 

5.  $111.42  for  4  years  2  months,  at  5%  ? 

6.  $19000  for  2  y.  4  mo.,  at  7^%  ?  Ans.  $3236.33^ 

7.  $6600  for  3  y.  6  mo.  20  da.,  at  (5%  ?  Ans.  $1408. 


SIMPLE   INTEREST.  195 

8.  What  is  the  interest  of  $750  from  Jan.  9,  1869,  to  Nov. 
9,  1870  ?  Ans.  $82.50. 

9.  What  is  the  amount  of  $3350  for  5  y.  9  mo.,  at  8%  ? 

10.  What  is  the  amount  of  $1242  from  July  3,  1868,  to  Jan. 
18,1870?  ^ns.  $1356.89. 

SIX  PER  CENT.   METHOD. 
470.  The  Interest  on  any  principal,  at  6%, 
For  12  months,  or  1  year,    is  .06  of  the  principal. 
"        2  months, 

"        1  month, 

j  month,    "  .001 
j^  month,''  .000^ 
^   month,  "  .000^ 


"        6  days, 
"        3  days, 
1  day. 
Hence,  the  following 


7  year,      "  .01 


^,  year,    "  .005 


30 


471.  Principles. — 1.  The  interest  at  6  per  cent,  for  any  num- 
ber of  months  is  equal  to  one  half  as  many  hundredths  of  the 
principal  as  there  are  inonths ;  and 

2.  The  interest  at  6  per  cent,  for  any  number  of  days  is  equal 
to  one  sixth  as  many  thousandths  of  the  principal  as  there  are 
days. 

WBITTEN  EXEMCTSES. 

472.— Ex.  1.  What  is  the  interest  of  $576  for  1  y.  7  mo.,  at 
fit/.? 


nol         SoXiTTTiON. — 1  y.  7  mo.  are  19  months. 

'2  Since  the  interest  at  6  %  is  one  half  as  many  hundredths 

5 184-        ^^  *^^®  principal  as  there  are  months  in  the  time,  it  must  be 

Qgg        one  half  of   19  hundredths,  or  .09|,  of  $576,  which  is 

$54.72. 


$54.72 


2.  What  is  the  interest  of  $950  for  2  y.  8  mo.,  at  6%  ? 

3.  What  is  the  interest  of  $420  for  3  y.  4  mo.,  at  7%  ? 


196  SIMPLE  INTEREST. 

4  What  is  the  interest  of  $455  for  2  y.  6  mo.  12  da.,  at  7%  ? 

$455 

152  Solution. — 2  y.  6  mo.  12  da.  are  30.4  mo.    Since 

the  interest  at  Q'^'o  is  one  half  as  many  hundredths 
of  the  principal  as  there  are  montlis  in  the  time, 
2275  it  must  be  one  half  of  30.4  hundredths,  or  .152, 

455  of  $455,  which  is  $69,160. 

r  )^PQ  1RC)  "^  ^  interest  is  \  more  than  6  fo  interest ;  hence, 

OJ^OJ.IOU  $69,160  plus  1  of  $69,160,  or  $80,687,  is  the  in- 

terest required. 


11.5266  + 


$80,687 

5.  What  is  the  interest  of  $940  for  33  da.  at  6%  ? 


'005-  Solution. — Since  the  interest  at  6%  is  one  sixth  as 

~  many  thousandths  of  tlie  principal  as  there  are  days  in 

Jf-i  UU  fijg  time,  it  must  be  one  sixth  of  33  thousandths,  or  .005|, 

470  of  $940,  which  is  $5.17. 


$5.17 

6.  What  is  the  interest  of  $756  for  8  mo.,  at  6%  ? 

An8.  $30.24. 

7.  What  is  the  interest  of  $631.20  for  11  mo.,  at  8%  ? 

473,  Rules  for  Interest  by  the  Six  Per  Cent.  Method.— i.  Mul- 
tiply the  principal  by  one  half  as  many  hundredths 
as  there  are  months,  or  by  one  sixth  as  many  thou- 
sandths as  there  are  days  in  the  time,  and  the  result 
will  be  the  interest  at  6  per  cent. 

^.  For  interest  at  any  other  rate  than  6  per  cent., 
increase  or  diminish  the  interest  at  6  per  cent,  by  such 
a  paH  of  itself  as  will  mahe  the  required  interest. 

PROIiLEMS. 

Wliat  is  the  interest  of — 

1.  $38.60  for  6  mo.  24  da.,  at  5%  ? 

2.  $1090  for  14 da.,  at  6%  ?  Arts.  $2.54. 

3.  $400.50  for  7  mo.  6  da.,  at  7%  ? 

4.  $5000  for  63  da.,  at  9%  ?  An^.  $78.75. 


SIMPLE  INTEREST.  197 

5.  $342  for  93  da.,  at  7%  ? 

6.  S1200  for  1  mo.  21  da.,  at  6%  ?  Ans.  $10.20. 

7.  $1560  for  1  y.  8  mo.,  at  7%  ? 

8.  $1920  for  2  y.  3  mo.,  at  5%  ?  Ans.  $216. 

9.  $500  from  January  15  to  December  2,  at  72%  ? 

10.  What  is  the  amount  of  $1345  from  April  9,  1870,  to 
September  5,  1871,  at  7%  ?  $1477.59. 

11.  What  is  the  amount  of  $3000  from  June  11,  1870,  to 
Aug-ust  17,  1871,  at  8%  ?  Ans.  $3284. 

SPECIAL   METHODS   FOR    DAYS. 

474-.  Tlie  Interest  of  any  principal  at  6%  for  2  months,  or 
60  days,  is  one  hundredth  of  the  principal.  Hence  the  fol- 
lovring — 

475.  Principle. — The  interest  of  any  jyrindpal  at  Q'fo  for 
any  number  of  days  is  as  many  hundredths  of  the  jmncipal  as 
60  is  contained  times  in  the  number  of  days. 

WMITTEW   JEXERCISJES. 

476.— Ex.  1.  What  is  the  interest  of  $240  for  93  da.,  at  6%  ? 

Solution. — TJie  interest  at 
Int.  of  $240  for  60  da.  =  $2.40      ^%    for  60  days  is   one  hun- 

Q6)      drcdtli    of   the    principal,   or 
~^—       $2.40. 

/^U  Since  for   any   number  of 

2160         days  it  is  as  many  hundredths 

fiO)  '^'P'P'^  ^0  ■     ^^  *^^®  principal  as  60  is  con- 

— — tained  times  in  the  number  of 

$3.72      days,  for  93  da.  it  must  be  ^V 
of  93  times  $2.40,  or  $3.72.  Or, 

Solution. — Since    the    in- 

$24-0  =  Principal.  terest  for  60  days  is  $2.40,  for 

$2.40  =  Int.  for  60  da.  ^0  days,  or  \  of  60   days,  it 

2  20=       "'      30  da.         """"^  ^^  ^  °^  ^'^■^^'  °''  ^^•"^' 


.12=       "  3  da. 


and  for  3  days,  or  y^y  of  30  days, 
it  must  be  ^V  of  $1.20,  or  $.12. 


>y2  =        "         Q^  da.  '^^Q  sum   of   these   results   is 

$3.72,    which    is    the   interest 
required. 


17  « 


198  SIMPLE  INTEREST. 

2.  What  is  the  interest  of  $120.60  for  11  da.,  at  6%  ? 

3.  What  is  the  interest  of  $500  for  123  da.,  at  5%  ? 

Ans.  $8.54. 

477.  Rule  for  Interest  by  Special  Method  for  Days.— i,  i2e- 
TYhove  the  decimal  point  in  the  principal  tivo  orders 
to  the  left,  for  the  interest  at  6  per  cent,  for  60  days. 

For  the  interest  for  any  other  number  of  days, 
multiply  the  interest  for  60  days  by  the  number  of 
days,  and  divide  by  60 ;  or,  talce  any  convenient 
multiples  or  aliquot  parts  of  the  interest  for  60 
days. 

PROBLEMS. 

What  is  the  interest  of — 

1.  $318.20  for  36  da.,  at  6%  ?  Ans.  $1.91. 

2.  $415  for  19  da.,  at  6%  ? 

3.  $31.25  for  16  days,  at  6%  ?  Aiis.  $.08. 

4.  $1120  for  153  days,  at  7%  ? 

5.  $6000  for  8  days,  at  12%  ?  Ans.  $16.00. 

6.  $311.50  for  35  days,  at  6%  ?  Ans.  $1.82. 

7.  $65.20  for  130  days,  at  7%  ? 

8.  Find  the  amount  of  $17000  for  75  days,  at  8%. 

CASE    II. 

Principal,  Interest  and  Time  Given,  to  Find  the  Rate. 

478.— Ex.  1.  At  what  rate  will  $576  earn  $86.40  in  2  years 
6  months  ? 

w  3       >r    -r-  -T-        at  1^  for  2^  years  is  5!l4.40. 

$86.40 -^$14.40  =  6  ''^i"<=e  the  principal  at  1%  in  2\ 

years  earns  $14.40,  to  earn  $86.40  it 
nuist  be  at  as  many  per  cent,  as  $86.40  is  times  $14.40,  or  at  6%. 

2.  At  what  rate  will  $1450  earn  $217.50  in  2  years  6 
months  ? 

479.  Rule  for  Finding  the  Rate  of  Interest.  -i)^z;z^e  the  given 
interest  by  the  interest  of  the  principal  for  the  given 
time  at  1  per  cent. 


SIMPLE  INTEREST.  199 

PltOJiLJEMS. 

At  what  rate  will — 

1.  $1760  earn  $246.40  in  2  years?  Am.  7%. 

2.  $110.25  gain  $11.02i  in  1  year  8  months?       Am.  6%. 

3.  $6600  gain  $1412.40  in  3  years  6  months  20  days? 

4.  $19000  gain  $3236.33^  in  2  years  4  months? 

Ans.  7j\%. 

5.  At  what  rate  will  $100  double  itself  in  16  years  8 
months?  Am.  6%. 

6.  At  what  rate  will  any  principal  double  itself  in  12|^  years? 

7.  If  you  should  pay,  January  22,  1871,  $735,  as  principal 
and  interest  for  $700  borrowed  July  22,  1870,  what  would  be 
the  rate?  Am.  10%. 

C.A.SE   III. 

Pi-incipal,  Rate,  and  Interest,  or  Amoimt  Given,  to  Find  the  Time. 

480.— Ex.  1.  In  what  time  will  $576  gain  $86.40,  interest 
being  at  6%  ? 

$576  X.06  =  $34.56  Solution.— The  interest  of  $576  for  1 

year,  at  6%,  is  $34.56. 
$86.4-0  -^  $34-56  =  2-         Hence,  it  must  require  as  many  years 

to  gain  $86.40  interest  as  $86.40  is  times 
$34.56,  or  '2\  years,  which  are  2  years  6  months. 

2.  In  what  time  will  $400  amount  to  $435,  at  7  %  interest  ? 

$435 —  $400  =  $35        Solution.  —  The  amount  $435  less    the 

(Ji(Qo        principal  $400  is  $35,  which  is  the  interest. 

X.(J7  —  ^^8  rpj^g  interest  of  $400  for  1  year,  at  7%,  is 

^$28=lj  $28. 

*  Hence,  it  must  require  as  many  years  to 

gain  $35  interest,  or  for  $400  to  amount  to  $435,  as  $35  is  times  $28,  or 
1;[  years,  wliich  are  1  year  3  months. 

3.  In  Vthat  time  will   $650  gain  $55.25,  interest  being  at 

6%  ? 

481.  Rule  for  Finding  the  Time  that  a  Sum  has  been  at  Interest. — 

Divide  the  given  interest  by  the  iiiterest  of  the  princi- 
pal for  one  year  at  the  given  rate. 


200  SIMPLE  INTEREST. 

mo  BL  JEMS. 

In  what  time  will — 

1.  $7000  gain  $588,  at  7%  interest  ?       A^is.  1  y.  2  mo.  12  d. 

2.  S6000  gain  $355,  at  5%  interest? 

3.  $100  double  itself,  at  7%  interest?      Ans.  14 y.  3f  mo. 

4.  $1250  amount  to  $1253.12^  at  6%  interest? 

Atis.  15  days. 

5.  Any  principal  double  itself,  at  8  %  interest  ? 

6.  $1828.60  amount  to  $2331.33f,  at  10%  interest? 

Ans.  2  y.  9  mo. 

Time,  Rate,  and  Interest,  or  Amount  (Jiren,  to  Find  the  Principal. 

482. — Ex.  1.  What  principal  will  earn  $86.40  in  2  y.  6  mo., 
at  6%  ? 

Soi^rTiON.— The  interest  of  $1  for  2 
■  ^1  X    Ofi  X  S-  =  ^  15      years  6  months,  at  6^^,  is  $.15. 

^  '  Hence,  it  must  require  as  many  dollars 

$86.40 -^$.15  =  576        to  earn  $86.40  as $86.40  is  times  $.15,  or 

$576. 

2.  What  principal  will  amount  to  v435  in  1  year  3  months, 
at  7%  ? 

.Solution. — The  amount   of  $1    for   1 
$1  +  $.08-  =  $1.08 J        year  3  months,  at  7f,,  is  $1.08}. 

Hence,  it  mu.st  require  as  many  dollars 
$435  -^  $1. 08 J  =  400      to  amount  to  $435  as  $435  is  times  $1 .08 1, 
or  $400. 

3.  What  principal  will  gain  $45.24^  in  3  years,  at  6%  ? 

Ans.  $251.80. 

483.  Rule  for  Finding  the  Principal  that  has  been  at  Interest— 

Divide  the  given  interest  or  aniojvnt  by  the  interest  or 
amount  of  $1  for  the  given  time  and  rate. 

riton  T.r.MS. 

What  principal  will — 

1.  Gain  $18.75f  in  90  days,  at  6%  ?  Ans.  $1250.50. 

2.  Gain  $246.40  in  2  years,  at  7%  ? 


SIMPLE  INTEREST.  201 

3.  Gain  $352.50  in  1  year  2  months  3  days,  at  5%  ? 

Ans.  $6000. 

4.  Gain  $3236. 33|  in  2  years  4  months,  at  7y\%  ? 

5.  Amount  to  $355.60  in  1  year  7  months  10  days,  at  8%  : 

Ans.  $315. 

6.  Amount  to  $200  in  14  years  3f  months,  at  7%  ? 

Ans.  $100. 


TEST    QUESTIONS. 

484. — 1.  What  is  Interest?  What  is  the  principal ?  The  amount? 
The  rate  ? 

2.  What  is  a  Legal  Rate  of  interest?  What  is  usury?  What  is  the 
rate  in  most  of  the  States,  when  no  rate  is  named  ?  What  are  the  rates 
in  other  States  ?     On  debts  due  the  United  States  ? 

3.  In  the  Computation  of  interest,  how  is  it  customary  to  reckon 
time?  In  finding  the  ditierenee  between  dates?  How  far  in  the  pro- 
cesses of  reckoning  interest  may  partial  results  be  carried?  What  is 
sufhcient  when  there  are  mills  in  the  answer  ? 

4.  What  is  Simple  Interest?  To  what  is  simple  interest  equal? 
What  is  the  rule  for  the  general  method  of  computing  interest  ? 

5.  To  what  is  interest  at  Qfo  for  any  number  of  months  equal?  For 
any  number  of  days?  What  is  the  rule  for  the  6^  method  of  com- 
puting interest? 

6.  What  part  of  the  principal  is  the  interest  at  6^  for  2  months? 
How  do  you  most  readily  find  the  interest  at  6^^  for  60  days?  For  any 
other  number  of  days  ? 

7.  When  the  principal,  interest  and  time  are  given,  how  do  you  find 
the  rate?  When  the  principal,  rate  and  interest,  or  amount,  are  given, 
how  do  you  find  the  time?  When  the  time  rate  and  interest  are 
given,  how  do  you  find  the  principal  ? 

8.  What  is  Percentage?  To  what  is  the  percentage  of  a  number 
equal?     To  what  is  the  rate  equal ?     To  what  is  the  base  equal ? 

9.  To  what  is  Profit  and  Loss  equal?  What  are  the  rules  for  profit 
or  loss  ? 

10.  What  is  the  base  of  Commission?  The  amount?  The  net  pro- 
ceeds ?     What  are  the  rules  for  commission  ? 

11.  How  do  property  and  life  insurance  differ?  What  are  the  rules 
for  insurance  ? 


202  FAHTIAL  PAYHENTS. 

SECTION    XLIX. 
PARTIAL  PAYMENTS. 

485.  A  Note,  or  a  Promissory  Note,  is  a  written  promise  to 
pay  a  certain  sum  of  money  for  value  received. 

486.  The  Maker  of  a  note  is  the  party  who  signs  it. 

487.  The  Payee  of  a  note  is  the  party  to  whom,  or  to  whose 
order,  it  is  to  be  paid. 

488.  An  Indorser  of  a  note  is  a  party  who  writes  his  name 
upon  the  back  of  a  note  or  other  obligation,  to  transfer  it  or 
to  guarantee  its  payment. 

489.  The  Face  of  a  note  is  the  sum  named  in  it. 
The  number  of  dollars  should  be  written  in  wards. 

490.  A  Time  Note  is  one  made  payable  at  a  specified  time. 
When  no  time  for  its  payment  is  specified,  the  note  is  due  on  demand. 

491.  Three  days,  called  Days  of  Grace,  are  usually  allowed 
for  the  payment  of  such  a  note. 

Thus,  a  note  payable  30  days  after  date  is  really  due  on  the  last  day 
of  grace,  or  33  days  after  date. 

492.  A  Negotiable  Note  is  one  so  made  that  it  can  be  sold  or 
transferred. 

FOEM  OF  A  NEGOTIABLE  NOTE. 

fQOOj^Q'  g^_  Louis,  January  4,  187L 

For  value  received,  I  promise  to  pay  to  the  order  of 

50 

Andrew  Hale  Eiglit  Hundred  Sixty  ^  Dollars,  on 

demand,  ivith  interest. 

Daniel  Wright. 

Andrew  Hale  can  transfer  this  note  by  sim])ly  writing  his  name  on  its 
back,  or  he  may  transfer  it  to  John  Jones  by  writing  upon  its  back, 

Pay  to  the  order  of  John  Jones. 

Andrew  Hale. 


PARTIAL    PAYMENTS.  203 

493.  Partial  Payments  are  part  payments  of  notes  or  other 
obligations  bearing  interest, 

494.  Indorsements  of  partial  payments  are  statements  of 
the  payments  on  the  back  of  the  instrument. 

495.  The  Supreme  Court  of  the  United  States,  and  most  of 
the  States,  adopt  for  partial  payments  a  rule  based  upon  the 
following : 

496.  Principles. — 1.  Payments  must  he  applied  first  to  the 
discharge  of  interest  due,  and  the  balance,  if  any,  toward  the 
discharge  of  the  principal. 

2.  Interest  must  not  be  added  to  the  principal  to  draw 
interest. 

3.  Interest  must  accrue  only  on  unpaid  principal. 

WJtITTEN  EXEMCISES. 

497.— Ex.  1.  A  note  for  $2000  was  given  July  1,  1870. 
Upon  it  was  paid,  as  by  indorsement,  January  1,  1871,  $260; 
July  1,  1871,  $50;  and  May  1,  1872,  $96.  Required  the 
balance  due  November  25, 1872. 

Solution. 
Principal,  $2000.00 

Int.  to  Jan.  1,  1871,  6  mo.,  *"               60.00 

Amount,  $2060.00 

First  payment,  Jan.  1,  1871,  260.00 

New  principal,  $1800.00 

Int.  to  May  1,  1872,  16  mo.,  I44.OO 

Amount,  $1944.00 

Second  payment,  July  1,  1871,  $50 

Third  payment.  May  1,  1872,  96        I46.OO 

New  principal,  $1798.00 

Int.  to  Nov.  25,  1872,  6.8  mo.  61.  IS 

Amount  or  balance  due,  $1859. IS 


204  PARTIAL   PAYMENTS. 

498.  Rule  of  the  Supreme  Court  of  the  United  States  for  Partial 
Payments.— i^^j/it^  the  cunount  of  the  principal  to  the 
time  when  the  payment,  or  the  sum  of  the  payments, 
equals  or  exceeds  the  interest  due,  and  subtract  the 
payment  or  the  sum  of  the  payments. 

Regard  the  remainder  as  a  neiv  principal,  and 
proceed  as  before. ' 

ritOBLJSMS. 

1.  Find  the  amount  due  December  29,  1871,  at  7%,  upon 
a  note  for  $960,  dated  Albany,  N.  Y.,  March  11,  1869;  and 
on  which  there  has  been  paid,  November  1, 1870,  $63.52 ;  and 
April  17,  1871,  $70.60. 

2.  Find  the  balance  due  April  1,  1872,  on  the  following 
note: 

$500.  Harrisbitrg,  May  16,  1869. 

For  value  received,  on  demand,  I  promise  to  pay 

Clia^rles   Berger,    or   bearer.  Five  Hundred   Dollars, 

with  interest,  ivithout  defalcation. 

John  Hofland 

Indorsements:  Jfov.  22,  1869,  received  Forty-five  Dol- 

50 

lars;  May  28,  1870,  received  Seventy  j^  Dollars. 

Ans.  $460.47. 

3.  Find  the  balance  due  July  25,  1872,  at  6%,  on  the  fol- 
lowing note : 

$6000.  Providence,  January  1,  1870. 

On  demand,  for  value  received,  I  promise  to  pay 
the  bearer  Six  Thousand  Dollars,  with  interest. 

James  D.  Mowry. 

Indorsements:  Jvly  1,  1870,  received  One  Tlwusand 
Dollars;  May  1.  1872,  received  Three  Thousand  Dol- 
lars. 


PARTIAL  PAYMENTS.  205 

4.  Find  the  balance  due  August  1,  1870,  at  8%,  on  the  fol- 
lowing note : 

$4000.  Richmond,  June  5,  18G9. 

/  proiyiise  to  pay,  to  the  order  of  Andrew  L.  Brown, 
Four  TliousaJid  Dollars,  with  interest.  Value  re- 
ceived. 

H.  M.  Reeves. 

Indorsements:  August  17,  1S69,  received  One  Thou- 
sand Bollars;  January  29,  1870,  received  One  Hun- 
dred Dollars.  ^^s^  $3198.91. 

499.  Business  men  often  settle  notes  and  interest  accounts, 
running  not  more  than  a  year,  by  the  following,  called — 

500.  The  Merchants'  Rule  for  Partial  Payments.—i^^Mt^  the 
amount  of  the  principal  at  the  time  of  settlement. 

Find  the  amount  of  each  payment  from  the  time 
itivas  made  until  settlement;  and  from  the  amount 
of  the  principal  subtract  the  amounts  of  the  pay- 
ments. 

In  mercantile  accounts,  settlements  are  made,  according  as  the  custom 
may  be,  either  at  the  end  of  the  civil  year  or  at  the  end  of  some  under- 
stood number  of  months. 

PltOTiT.EBIS. 

1.  A  note  for  $1500,  on  demand  with  interest,  dated  Jan- 
uary 1,  1870,  had  paid  on  it.  May  1,  1870,  $800,  and  May  16, 
1870,  $300.  How  much  was  due  at  the  end  of  the  first  six 
months,  interest  being  at  7%  ?  Ans.  $440.54 

2.  Find  the  balance  due  at  the  end  of  the  year  on  a  note  for 
$650,  given  April  9,  1870,  on  which  has  been  paid,  June  9, 
$150,  and  November  30,  $200.  Ans.  $322.33. 

3.  Find  the  balance  due  August  5,  1871,  on  a  note  for 
$1275,  given  September  30,  1870,  on  which  has  been  paid, 
December  5,  1870,  $55 ;  January  9, 1871,  $760  ;  June  3, 1871, 
8400. 

18 


206  PRESENT   WORTH  AND  DISCOUNT. 

SECTION   L. 
FEi:SEJVT  WORTH  A^D  DISCOUNT. 

501.  Discount  is  a  sum  deducted  from  a  price  or  debt.  Its 
computation  may  have  reference  to  time,  or  not,  according  to 
the  kind  of  deduction  understood. 

COMMEECIAL  DISCOUNT. 

502.  Commercial  Discount  is  a  per  cent,  deducted  from  a 
price  or  from  the  face  of  a  bill,  without  reference  to  time. 

503.  The  Net  Price  of  an  article  is  the  selling  price  less  the 
discount. 

504.  The  Cash  Talue,  or  Net  Proceeds,  of  a  bill  is  its  face 
less  the  discount. 

505.  The  Base  of  commercial  discount  is  the  selling  price, 
or  the  face  of  a  bill. 

506.  The  Rate  is  the  rate  per  cent,  of  deduction. 

507.  The  Discount  is  the  percentage  of  the  deduction. 

508.  Rules  for  Commercial  Discount— MuUiplj/  the  selling 
price,  or  the  face  of  the  hill,  hy  the  rate  per  cent,  of 
deduction,  and  the  result  will  be  the  coimnercial 
discount. 

Subtract  frorn  the  selling  price,  or  from  tlie  face  of 
the  bill,  the  coimnercial  discount,  and  tlie  result  will 
be  the  net  price,  cash  value,  or  net  proceeds. 

PliOBLEMS. 

1.  What  is  the  net  cash  price  of  flour  invoiced  at  $12.40 
per  barrel,  on  30  days'  time,  or  5%  oflT  for  cash  ? 

A)).<t.   811.78  per  barrel. 
'  2.  What  is  the  cash  value  of  goods  amounting  as  by  bill  to 
$1565,  discount  off  being  20%,  and  10%  off  for  cash? 

Am.  $1126.80. 


PRi:SENT   WORTH  AND  DISCOUNT.  207 

3.  I  sold  a  bill  of  books  amounting  to  $980,  taking  10%  off 
for  60  days,  and  5%  off  for  cash.  What  was  the  cash  value 
of  the  bill  ?  ^ns.  $837.90. 

TRUE  DISCOUNT. 

509.  The  Present  "Worth  of  a  debt  payable  at  a  future  time, 
without  interest,  is  such  a  sum  as,  being  placed  at  interest, 
will  amount  to  the  given  debt  when  it  becomes  due. 

Thus,  $100  is  the  present  worth  of  $106  due  1  year  hence,  at  6%. 

510.  The  True  Discount  is  the  difference  between  the  face 
of  the  debt  and  the  present  worth ;  or, 

It  is  the  interest  on  the  present  worth  for  the  time  interven- 
ing between  the  payment  of  the  debt  and  the  time  of  its  be- 
coming due. 

511.  The  present  worth  corresponds  to  the  principal,  the 
debt  to  the  amount,  and  the  discount  to  the  interest.  (Art.  483.) 
Hence,  the  following — 

512.  Rules  for  True  Discount.— i.  Divide  the  face  of  the 
debt,  or  the  given  sum,  by  the  amount  of  $1  for  the 
given  time  and  rate,  and  the  result  will  be  the  present 
worth. 

2.  Subtract  the  present  ivorth  froin  the  given  sum, 
and  the  result  will  he  the  true  discount. 

PJiOBLEMS. 

1.  What  is  the  present  worth  and  the  true  discount  of  $408, 
payable  in  4  months,  at  6%  ? 

Ans.  Present  worth,  $400 ;  true  discount,  $8. 

2.  What  sum  of  ready  money  is  equivalent  to  $3350,  due  1 
year  8  months  hence,  when  money  is  worth  7%  ? 

3.  What  is  the  true  discount  on  $625,  due  3  months  hence, 
at  8%  ?  Ans.  $12,251 

4.  If  I  pay  a  debt  of  SI 200,  2  years  6  months  before  it  is 
due,  what  discount  should  be  made,  money  being  worth  6%  ? 

5.  I  am  offered  goods  for  $4050  cash,  or  for  $4253.75  on  3 
months.  Which  is  the  better  offer,  and  how  much  better  is  it, 
if  money  is  worth  10%  ?  Ans.  Cash,  by  $100. 


208 


BANKING. 


fe%;M^ 


SECTION   LI. 

513.  A  Bank  is  an  institution  established  by  law  for  receiv- 
ing deposits,  loaning  money,  or  issuing  notes  or  bills  to  circu- 
late as  money. 

514.  A  Deposit  is  money,  or  its  equivalent,  entrusted  to  the 
care  of  a  bank. 

515.  A  Check  is  a  written  order,  or  request,  for  money,  ad- 
dressed to  a  bank  by  a  person  having  a  deposit. 


No.  13, 


Pay  to^ 


FORM   OF  A   BANK   CHECK. 

Hartford,  January  3,  1872. 
Traders'  National  Bank. 


.Henry  Brown,- 


.07'  hearer, 


$6000{l. 


^Six  Thoivsand- 


Jqq  Dollars. 


J.  F.  Pratt. 


516.  Bank  discount  is  the  interest  deducted  from  the  face 
of  a  note,  for  the  payment  of  its  proceeds  before  the  note 
becomes  due. 

517.  The  Proceeds  or  Avails  of  a  note  discounted,  are  the 
face,  or  amount,  of  the  note,  less  the  discount. 


BANKING.  209 

FOEM  OF  A  NOTE  PAYABLE  AT  A  BANK. 

r_^.  New  Orleans,  May  IG,  1871. 

Minety   days  after  date,  I  promise  to  pay  to  the 

order  of Jeaii  Rernond at   the  First 

National  Banh Seven  Hundred  Fifty- 

50 

five ^  Dollars.     Value  received. 

Paul  Ganot. 

518.  The  Maturity  of  a  note  is  the  time  on  which  it  is  legally 
due,  and  that  is  the  last  day  of  grace.  When  the  last  day  of 
grace  happens  on  Sunday  or  a  legal  holiday,  the  note  is  pay- 
able a  day  earlier. 

A  note  is  payable  or  nominally  due  at  the  time  specified  in  the  note. 
The  times  of  being  payable  and  of  maturity  are  usually  indicated  by 
writing  the  numbers  of  the  days  of  the  two  dates  with  a  line  between 
them.    Thus,  July  13/jg^  i871. 

A  bank,  on  discounting  a  note,  retains  the  discount,  holds  the  note  and 
pays  the  proceeds,  or  avails,  to  the  former  holder. 

519.  The  Term  of  Discount  is  the  number  of  days  from  the 
time  of  discounting  to  the  maturity  of  the  note. 

The  term  of  discount  is  called  the  time  to  run. 

When  a  note  is  not  paid  at  maturity,  a  written  notice,  called  a  protest, 
should  be  made  out  and  sent  to  the  indorser  or  indorsers  by  an  officer 
known  as  a  Notary  Public. 

In  order  to  hold  the  indorsers  for  the  payment  of  the  note,  this  should 
be  done  on  the  last  day  of  grace. 


CASE   I. 

The  Face  of  a  Note,  Time  and  Rate  giyen,  to  Find  tlie  Bank 
Discount,  or  Proceeds. 

520. — Ex.  1.  Required  the  bank  discount  and  proceeds  of  a 
note  for  $800,  for  3  months,  dated  Wilmington,  Del.,  August  5, 
and  discounted  September  6,  at  6%. 


210  BANKING. 

Solution. — Ma- 
Int  of  ^800  for  60  da.  =  $8.00  turity,  or  3  months 

Q^  plus  3  days  of  grace 

^  ^  \ ^  -„  .   ^^  from    August  5,    is 

60)$504.00       _  Novembers.     The 

^ 8 .4-0 , Discount,    term  of  discount,  or 

tlie  number  of  davs 

$800  -  $8.40  =  $791.60,  Proceeds.      from    September "  (i 

to  November  8,  is  63 
days. 
Interest  of  $800  for  60  days  at  6%  is  .01  of  $800,  or  $8.00,  and  for  63 
days  at  6;^  is  |f  of  $8,  or  $8.40,  which  is  the  required  discount. 

The  face  of  the  note  less  the  discount  is  $800  — $8.40,  or  $791.60; 
which  is  the  proceeds. 

2.  What  is  the  bank  discount  of  a  note  for  $500,  at  30  days, 
at  6%  ?  Am.  §2.75. 

521.  Rules  for  Bank  Discount.— i.  Find  the  interest  on  the 
face  of  the  note,  at  the  given  rate,  for  the  term  of 
hanh  discount,  and  the  result  ivill  be  the  hanh  dis- 
count. 

2.  Subtract  the  hanh  discount  from  the  face  of  the 
note,  and  the  difference  will  he  the  proceeds. 

PJtOB  1.EMS. 

1.  I  bought  a  horse  for  $250,  and  gave  my  note  at  90  days. 
How  much  ready  money  did  he  cost  me,  discount  being  at  8%  ? 

Ans.  $244.83. 

2.  What  are  the  proceeds  of  a  note  for  $1220,  due  in  4 
months,  discount  at  7  %  ? 

3.  What  is  the  bank  discount  of  a  note  for  $193,  dated 
July  1,  payable  in  60  days,  with  interest  at  5%,  discounted  15 
days  after  date,  at  6%  ?  Ans.  $1,56. 

4.  Find  the  time  of  becoming  due,  term  of  discount,  bank 
discount  and  proceeds  of  a  note,  dated  May  5,  1871,  for 
$1450.25,  payable  in  6  months,  and  discounted,  July  11, 
1871,  at  7%. 

Ans.  Due,  Nov.  'Vs ;  term  of  discount,  120  days; 
discount,  $33.84 ;  proceeds,  $1416.41. 


BANKING.  211 

5.  Find  the  time  of  becoming  due,  term  of  discount,  banlc 
discount  and  proceeds  of  the  following  note,  which  was  dis- 
counted August  8. 

$567 O'.  Eeading,  July  5,  1871. 

Jfinety  days  after  date,  I  promise  to  pay  to  the  order 
of  Luther  Getz  c^-  Co.,  Five  Thousand  Six  Hundred 
Seventy  Dollars,  at  the  First  Kational  Bank,  without 

defalcation.     Value  received. 

George  Hartnvan. 

Ans.  Due,  Oct.  ^/q  ;   term  of  discount,  59  days ; 
discount,  $55.76 ;  proceeds,  $5614.24. 


CASE   II. 

Proceeds  or  Bank  Discount,  Time  and  Rate  given,  to  find  the  Face 
of  the  Note. 

523. — Ex.  1.  The  proceeds  of  a  note  discounted  for  90  days, 
at  6%,  are  $787.60.     Required  the  face  of  the  note. 

Solution.  —  The 

Bank  dmount  of  $1  for  03  days  =  $.0155    \^^l^^^^^Z%tltl. 

Proceeds  of  $1  for  93  days=$.9845        The  proceeds  of 

$7 87. 60 ^.9845  =  $800         ^l  ^^^  ^3  days  are 

$1   less    $.0155,   or 
$.9845. 
Since  $.9845  is  the  proceeds  of  $1  for  the  given  time  and  rate,  $787.60 
must  be  the  proceeds  of  as  many  dollars  as  $787.60  is  times  $.9845,  or 


2.  For  what  sum  must  a  6  months'  promissory  note  be 
written,  that  the  proceeds,  at  7%  discount,  maybe  $484.75? 

Ans.  $502.64. 

523.  Rules  for  finding  the  Face  of  a  Note.— i.  Divide  H^e 
given  proceeds  hy  the  proceeds  of  $1  for  the  given 
time  and  rate.     Or, 

2.  Divide  the  given  discount  hy  the  discount  of  $1 
for  the  given  time  and  rate. 


212  BANKING. 

PROBLEMS. 

1.  The  bank  discount  on  a  certain  note  discounted  for  3 
months,  at  6%,  was  612.40.     What  was  the  face  of  the  note? 

Ans.  6800. 

2.  What  must  be  the  face  of  a  note  in  order  that  when  dis- 
( ouuted  for  60  days,  at  6%,  its  proceeds  shall  be  6394? 

3.  The  proceeds  of  a  30  days'  note,  discounted  at  8^,  were 
61335.63.     What  sum  will  pay  the  note  at  maturity  ? 

Am.  61345.50. 

4.  For  what  sum  must  a  note  be  made  to  obtain  6650  from 
a  bank,  for  57  days,  at  6  %  discount  ? 

5.  For  what  sum  must  a  note,  dated  March  9,  on  4  months, 
be  drawn,  so  that  if  discounted  at  5%,  May  9,  it  shall  yield 
61090.90?  .i^is.  61100.68. 


TEST   QUESTIONS, 

524. — 1.  What  is  a  Promissory  Note  ?  Who  is  the  maker  ?  The 
payee?    An  indorser?     What  is  the  face  of  a  note? 

2.  What  is  a  Time  Note?  What  are  days  of  grace?  When  is  a 
time  note  really  due  ? 

3.  ^Vhat  is  a  Negotiable  Note  ?  How  may  a  negotiable  note  be 
transferred  ? 

4.  What  are  Partial  Payments?  Indorsements?  What  is  the 
United  States  rule?     The  merchants'  rule? 

5.  What  is  Discount  ?  Commercial  discount  ?  The  net  price  of  an 
article  ?     The  cash  value  of  a  bill  ? 

6.  What  is  the  Base  of  commercial  discount  ?  Of  what  is  the  dis- 
count the  percentage?  What  is  the  rule  for  computing  commercial 
discount  ? 

7.  What  is  the  Present  Worth  of  a  debt  ?  What  is  true  discount  ? 
What  is  the  rule  for  the  computation  of  true  discount  and  present  worth? 

8.  What  is  a  Bank  ?  A  deposit?  A  check  ?  Bank  discount  ?  Wliat 
are  the  proceeds  or  avails  of  a  note  discounted  ? 

9.  What  is  the  Maturity  of  a  note?  When  is  a  note  payable?  What 
is  done  in  discounting  a  note?     What  is  the  term  of  discount? 

10.  What  is  the  Rule  for  finding  the  bank  discount  or  proceeds  of  a 
note?  Wlien  tlic  proceeds  or  bank  discount,  time  and  rate  are  given, 
liow  is  tlie  face  of  llie  note  found? 


ANNUAL    INTEREST.  213 

SECTION    LII. 
AJ^JfUAL  INTEREST. 

525.  iVimual  Interest  is  simple  interest  on  the  principal  and 
on  each  year's  interest  from  the  time  of  its  accruing,  till  settle- 
ment. 

526. — Ex.  1.  How  much  interest  is  due  on  a  debt  of  $500, 
at  6%  annual  interest,  at  the  end  of  3  years  6  months)? 

Solutions. 
$500y..06~  $30  =  each  year's  int.  on  the  principal. 


Or, 


X.21  =  $105.00  =  simple  int.     "  "    for  3  y.  6  mo. 

X.J?5=  4-50=^  "  on  1st  year's  int.  for  2  y.  6  mo. 
X.09=  2.70=  "  on2d  "  "  for  1  y.  6  mo. 
X.03=  .90=       "on  3d    "      "  for  6  mo. 

$113.10  =   the  annual  interest 

X  .21  =$105.00=  simple  int.  on  theprin.for  3  y.  6 mo. 
X .27  =         8.10=  "       on  $30  for  4  y.  6  mo. 

$113.10=  the  annual  interest. 

In  the  second  solution  the  computation  is  abridged  by  finding  the 
interest  on  one  year's  interest  for  4  y.  6  mo.,  which  is  the  sum  of  the 
periods  for  which  the  yearly  interests  remained  unpaid. 

2.  What  is  the  amount  of  $720  for  2  years  8  months,  at  7% 
annual  interest  ?  Ans.  $862.63. 

527.  Rule  for  Annual  Xnieve^^X.— Compute  interest  on  the 
principal  for  the  entire  time,  and  on  one  yeai-'s 
interest  for  the  sum  of  the  periods  during  which 
the  yearly  interests  remain  unpaid. 

The  sum  of  these  interests  will  he  the  interest 
required. 


214  ANNUAL  INTEREST. 

PJtOBL  Ejrs. 

1.  What  is  the  annual  interest  for  3  years  on  a  debt  of 
SIOOO,  at  6%,  annual  interest? 

2.  How  much  is  due  May  1,  1870,  on  a  note  for  §350,  dated 
January  1,  1868,  with  interest  payable  annually  at  8%  ? 

A)is.  $419.06. 

3.  How  much  is  due  May  16,  1871,  on  a  note  for  §600, 
dated  March  3,  1868,  with  interest  payable  annually  at  Q%, 
provided  the  yearly  interest  has  been  promptly  paid  ? 

Ans.  $607.30. 
528.  When  Partial  Payments  have  been  made  on  notes  or 
other  obligations,  with  "interest   annually,"  by  the  law   of 
Vermont, 

Find  the  interest  on  each  payinent  to  the  time  the 
interest  on  the  principal  next  becomes  due,  and  find- 
ing the  amount,  apply  it— first,  to  liquidate  any 
interest  that  may  have  accrued  upon  unpaid  yearly 
interests;  secondly,  to  liquidate  yearly  interests  that 
may  have  become  due;  and  thirdly,  to  the  ecc- 
tinguishinent  of  the  principal. 

By  the  New  Hampshire  Rule,  payments  only  that  exceed  the 
interest  due  at  the  time  made  are  allowed  interest. 

FJlOHTiEMS. 

1.  Find  the  balance  due  May  1,  1871,  on  the  following  note, 
by  the  Vermont  and  by  the  New  Hampshire  Rules : 

$600.  Concord,  N.  H.,  May  1,  1868. 

On  demand,  I  promise  to  pay  to  the  order  of  James 
Upham,  for  value  received,  Six  Hundred  Dollars, 
with  interest  annually. 

Benjamin  Hill. 

Indorsements:  July  1,  1868,  received  $100 ;  August 
9,  1869,  received  $25;  Sept.  1,  1870,  received  $300; 
Jan.  1,  1871,  $110. 

Am.  By  Vt.  liule,  S144.78 ;  by  N.  H.  Rule,  3147.43. 


CGMI'O  USD  INTEREST. 


215 


2.  Required  the  balance  due  December  3,  1871,  ou  a  note 
for  ^500,  dated  September  3,  1867,  bearing  interest  annually, 
by  the  New  Hampshire  Rule ;  and  indorsed,  April  19,  1869, 
received  ^63  ;  November  9,  1870,  received  $321.50. 

Ans.  $224.86. 


SECTION   LIII. 
COMPOUJ^D  IMTEEEST. 

529.  Compound  Interest  is  interest  on  interest  and  principal 

combined,  at  specified  intervals. 

Interest  may  be  compounded,  or  made  part  of  the  principal, 
annually,  semi-annually,  quarterly,  etc. 

530. — Ex.  1.  What  is  the  compound  interest  of  $744  for  2 
years  4  months,  at  5%  ? 


.05 


.20  Int.  for  1st  y. 


7U 


$781.20  Amt.  for  1st  y. 
.05 


Solution.  —  The 
amount  of  $744  for  the 
first  year  is  $781.20, 
which  we  make  a  prin- 
cipal for  the  second 
year. 

The  amount  at  com- 
pound interest  of  $744 
for  2  years  is  $820.26, 
which  we  make  a  prin- 
cipal for  the  remaining 
4  months  of  the  time. 

The  amount  at  com- 
pound interest  of  $744 
for  2  years  4  months  is 
$833.93  +. 

The  last  amount, 
minus  the  given  prin- 
cipal, is  $89.93,  which 
is  the  compound  inter- 
est required. 
2.  What  is  the  amount  at  compound  interest  of  $520  for  4 
years,  at  6%  ?  Am.  $656.49. 


0600  Int.  for  2d  y. 
781.20 
$820.26      Amt.forMy. 

.oii 


16710 
820.26 
$833.9310 
7U 
$89.93 


Int.  for  Jf  mo. 
Ami.  for  2  y.  4-  ^^o. 
Comp.  Int.  for  2y.  4-  '^^o. 


216  COMPOUND  INTEREST. 

531.  Rule  for  Compound  interest.— T^/zirZ  the  amoujvb  of 
the  given  priiici/jul  for  the  first  interval,  at  the  given 
rate. 

Make  that  amount  a  principal  for  the  second  in- 
terval, and  so  on. 

From  the  last  amount  subtract  the  given  principal, 
and  the  remainder  will  he  tlie  compound  interest. 

PnOBLIiMS. 

1.  What  is  the  compound  interest  of  83010'  for  3  years  3 
months,  at  7%  ?  .4?is.  $741.91. 

2.  What  is  the  interest  of  S9000  for  5  years  6  months,  in- 
terest compounding  at  3%,  semi-annually?     Ans.  $3458.09. 

3.  What  is  the  amount  of  8506.50  for  2  years  3  months, 
interest  compounding  at  \\%,  quarterly? 

4.  What  sum  will  pay,  January  1,  1872,  85000,  borrowed 
July  1,  1869,  at  7%,  compound  interest?        Ans.  85924.86. 

5.  Find  the  amount  of  8150.25,  at  compound  interest,  from 
May  16,  1870,  to  July  1,  1873,  at  6%. 

6.  Find  the  compound  interest  of  83000  from  October  3, 
1869,  to  April  3,  1872,  at  8%.  Ans.  8639.17. 


TEST  QUESTIONS. 
532. — 1.  What  is  Simple  Interest?    The  general  method  of  com- 
puting simple  interest  ? 

2.  What  is  Discount?  Commercial  discount?  How  do  true  dis- 
count and  bank  discount  differ?     What  are  the  rules  for  bank  discount? 

3.  What  are  Partial  Payments?  What  is  the  United  States  Eule 
for  computing  i)artial  payments  ? 

4.  What  is  Annual  Interest?  How  does  it  differ  from  simple  in- 
terest?    What  is  the  method  of  computing  annual  interest? 

5.  What  is  the  Rule  for  computing  partial  payments  by  the  Vermont 
Annual  Interest  Method?  How  does  the  New  Hampshire  Method 
differ  from  that  of  Vermont? 

G.  Wliat  is  Compound  Interest?  IIuw  does  it  differ  from  annual 
interest?     Wliat  is  the  rule  for  computing  compound  interest? 


REVIEW  PROBLEMS.  217 

SECTION   LIV. 

REVIEW   PROBLEMS. 

MENTAIj  exjemcises. 

533.— Ex.  1.  What  is  the  interest  of  ^15  for  4  years,  at  Q^/o  ? 
For  2  years,  at  5%  ? 

2.  What  is  the  interest  of  850  for  8  months,  at  8%  ?  For 
18  months,  at  10%  ? 

3.  What  is  the  interest  of  $300  for  1  month  20  days,  at  6%  ? 

Solution. — The  interest  of  $300  for  1  year  is  $18  ;  hence,  for  1  month, 
or  y^2  of  ^  year,  it  is  -^j  of  $18,  or  $l.j,  and  for  20  days,  or  f  of  a  month, 
it  is  t  of  %lh,  or  $1 ;  $U  and  $1  is  $2.50. 

4.  What  is  the  interest  of  8600  for  3  months  15  days,  at 
8%? 

5.  What  is  the  interest  of  $30  for  3  months  3  days,  at  7%  ? 

6.  In  what  time  will  8100  gain  818  interest,  at  6%  ?  $20 
interest,  at  5  %  ? 

7.  In  what  time  will  8200  gain  821  interest,  at  7%  ?  $26, 
at  8%  ? 

8.  In  what  time  will  860  gain  8.30  interest,  at  6%  ?  82.10, 
at7%? 

9.  What  is  the  interest  of  860  for  60  days,  at  6%  ?  For  93 
days,  at  6%? 

10.  In  what  time  will  a  given  principal  double  itself  at  1  %  ? 
At5%?    At6%? 

11.  In  what  time  will  85  gain  85,  at  8%  ?     At  10%  ? 

12.  At  what  per  cent,  will  8400  in  2  years  6  months  gain 
860  interest  ? 

Solution.— $400  in  2  years  6  months,  aX\%,  will  gain  $10;  hence,  it 
will  gain  $60  in  the  same  time  at  as  many  per  cent,  as  $10  is  contained 
times  in  $60,  or  6  per  cent. 

13.  At  what  per  cent,  will  8120  in  1  year  9  months  gain  821  ? 

14.  At  what  per  cent,  will  8100  in  2  years  3  months  gain  $18? 

15.  At  what  per  cent,  will  a  given  principal  double  itself  in 
20  years?     In  1G|  years? 

19 


218  JIEVII^W  PROBLEMS. 

16.  What  is  the  amount  of  $150  for  3  years  9  months, 
at  8%  ? 

17.  What  principal,  at  6%,  in  5  years  will  amount  to  $130? 

18.  What  sum  put  at  interest,  at  7%,  will  earn  for  me  $210 
yearly  ? 

19.  How  much  greater  is  the  annual  than  the  simple  interest 
ou  $500  for  4  years,  at  10%  ? 

20.  What  are  the  proceeds  of  a  oO-day  note  for  $400,  dis 
counted  at  12%  ? 

WltlTTElf  EXERCISES. 

534.— Ex.  1.  If  you  borrow,  June  12,  1870,  $500,  at  6% 
interest,  what  amount  will  you  owe  by  delaying  pa}Tiient  to 
July  15,  1871  ?  Am.  $532.75. 

2.  I  borrowed  $500,  June  12,  1870,  and  paid  July  15,  1871, 
as  principal  and  interest,  $532.75.  What  was  the  rate  of 
interest  ? 

3.  What  is  the  amount  of  $5000  for  2^  months,  at  1^%  per 
month?  Ans.  $5187.50. 

4.  What  sum,  put  at  interest  at  6%,  will  give  an  income  of 
$40  per  month  ?  Am.  $8000. 

5.  The  amount  of  money  I  have  at  interest  is  $7300.  It 
gives  me  exactly  $1  a  day,  each  common  year.  What  is  the 
rate  of  interest?  Am.  5%. 

6.  Chandler's  money,  at  6%  interest,  earns  for  him  $1140, 
and  Benton's,  at  8%,  earns  for  him  $60  more.  Which  has 
the  more  money,  and  how  much  more  ? 

Am.   Chandler,  $4000. 

7.  What  sum  must  a  father  invest  for  his  son  when  he  is  16 
years  8  months  old,  at  6%,  that  on  coming  of  age,  he  may 
have  $2100?  ^/(s.  $1666.66|. 

8.  What  was  due  November  12,  1870,  on  a  note  for  $350, 
<lated  jMay  4,  1869,  at  6%  interest,  and  on  which  are  indorsed, 
January  14,  1870,  received  $50,  and  June  13,  1870,  received 
$25? 

9.  I  bought  a  bill  of  goods  amounting  to  $4500  on  2  months' 
time,  or  5%  oflf  for  cash.     Should  I  have  gained  or  lost  by 


RA  TIO.  219 

borrowing  the  money  of  tlie  bank  on  a  60-clay  note,  discount 
being  at  6%  ? 

10.  If  a  legacy  of  $2400  is  left  me  on  the  3d  of  May,  to  be 
paid  on  the  Christmas-day  following,  what  is  its  present  value, 
money  being  worth  5%  ?  J.ws.  S2325.08. 

11.  How  much  greater  is  the  bank  than  the  true  discount 
on  $7400  due  4  years  8  months  hence,  without  grace,  at  7%  ? 

12.  How  much  more  is  the  compound  than  either  the 
annual  or  simple  interest  on  $5000  for  6  years,  at  6  %  ? 

Ans.  $22.60  more  than  the  annual  interest;  $292.60 
more  than  the  simple  interest. 


SECTION   LV. 
RATIO. 

535. — Ex.  1.  If  one  house  is  12  feet  high  and  another  24 
feet  high,  how  many  times  as  high  as  the  first  is  the  second  ? 

2.  What  part  of  $25  is  $5  ?     Of  $60  is  $3  ? 

3.  John  is  11  years  old,  and  his  uncle  is  33  years  old.  How 
does  his  age  compare  with  that  of  his  uncle  ? 

4.  What  part  of  33  years  is  11  years  ?  How  many  times  11 
years  are  33  years  ? 

5.  What  relation  has  1  to  6  ?     8  to  24? 

6.  What  is  the  relative  value  of  $15  compared  with  $3? 
$62  compared  with  $9  ? 

DEFINITIONS. 

536.  Ratio  is  the  relation  which  one  of  two  similar  num- 
bers bears  to  the  other  with  respect  to  value.  It  is  ascertained 
by  the  division  of  the  first  of  the  numbers  by  the  second. 

Thus,  the  ratio  of  $6  to  $3  is  2. 

537.  The  Terms  of  a  ratio  are  the  two  numbers  whose 
values  are  compared. 

The  first  term  is  called  the  Antecedent,  and  the  second  term 
is  called  the  Consequent. 


220  RATIO. 

538.  The  Sign  of  ratio  is  the  colon  (:),  -which  is  the  sign  of 
division  (h-)  with  the  line  between  the  dots  left  out. 

Thus,  7  :  8  denotes  the  ratio  of  7  to  8,  or  |. 

539.  A  Simple  Katio  is  a  ratio  each  term  of  which  is  a 
single  number. 

Thus,  9  :  3  is  a  simple  ratio. 

540.  A  Compound  Katie  is  a  ratio  formed  by  multiplying 
together  the  corresponding  terms  of  two  or  more  simple  ratios. 

Thus,  (6  :  2)  X  (5  :  3),  or  -  X  -^,  or     "       expresses  a  compound  ratio. 

541.  Principles. — 1.   The  tenuis  of  a  ratio  must  he  similar 
numbers. 

2.  The  ratio  of  tivo  numbers  is  the  quotient  of  the  antecedent 
divided  by  the  consequent. 

3.  Both  term^  of  a  ratio  may  he  multiplied  or  divided  by  the 
same  number  without  affecting  its  value. 

4.  A  compound  ratio  is  the  ratio  of  the  j^roduct  of  its  antece- 
dents to  the  product  of  its  consequents. 

EXERCISES. 

542.— Ex.  1.  Indicate  the  ratio  of  8  to  5.     Of  6  to  4. 

Ans.  8:5;  6:4. 

2.  Express  by  a  fraction  the  ratio  of  11  to  12.      Ans.  |^. 

3.  Find  the  ratio  of  75  to  25.     Of  19  to  6,     Am.  3  ;  ^. 

4.  Find  the  ratio  of  13  to  11 ;  of  15  to  31. 

5.  What  is  the  ratio  of  8  yards  to  2  feet  ?  Ans.  12. 

6.  If  the  antecedent  be  63,  and  the  consequent  9,  what  is 
the  ratio  ?  Am.  7. 

7.  If  the  ratio   be  7,  and  the  consequent  9,  what  is  the 
antecedent  ? 

8.  If  the  consequent  be  7,  and   the  ratio  9,  what  is  the 
antecedent  ? 

9.  If  the  ratio  be  7,  and  the  antecedent  63,  what  is  the 
consequent  ? 

10.  Express  the  ratio  of  ^  to  f . 


PROPORTION.  221 

11.  The  antecedents  of  a  compound  ratio  are  3,  8  and  10, 
and  the  consequents  are  8,  4  and  5.  What  is  the  compound 
ratio  ? 

12.  Express  the  ratio  of  a  field  6  rods  long  and  4  rods  wide, 
to  a  field  3  rods  long  and  2  rods  wide. 

13.  In  12  days,  8  men  can  do  how  many  times  as  much 
work  as  3  men  can  do  in  16  days  ? 


SECTION    LVI. 
FROFORTIOK. 

543.— Ex.  1.  What  is  the  ratio  of  2  to  1  ?     Of  10  to  5? 
Of  12  to  6? 

2.  What  two  numbers  have  the  same  quotient  as  10  ^-  5  ? 
As  18^6? 

3.  What  two  numbers  have  the  same  ratio  to  each  other  as 
18  has  to  6  ?     As  12  has  to  8  ? 

4.  When   2   oranges   cost   8   cents,  what  will   10   oranges 
cost? 

5.  AVhat  is  the  ratio  of  10  oranges  to  2  oranges?    Of  40 
cents  to  8  cents  ? 

6.  What  is  the  ratio  of  40  to  8  expressed  as  a  fraction  ?   Of 
14  :  3  expressed  as  a  fraction  ? 

7.  8  cents  are  y*  of  how  many  cents  ? 

8.  What  number  has  to  8  cents  a  ratio  equal  to  the  ratio 
of  10  to  2  ? 

9.  If  9  men  can  do  a  piece  of  work  in  25  days,  how  many 
men  can  do  the  same  in  5  days  ? 

10.  9  men  are  -^  of  how  many  men  ? 

11.  What  number  of  men  has  to  9  men  a  ratio  equal  to 
the  ratio  of  25  to  5  ? 

12.  12  is  ^  of  what  number?     12  is  to  what  number  as 
4  :  7? 

13.  27  dollars  are  ^-^  of  how  many  dollars  ? 

14.  To  what  number  has  27  dollars  a  ratio  equal  to  the 
ratio  of  54  to  6  ? 

19* 


222  PROPORTION. 

DEFINITIONS. 

544.  Proportion  is  an  equality  of  ratios. 
Thus,  4:2  =  6:3  expresses  a  proportion. 

545.  The  Sign  of  proportion  is  a  double  colon  ( :  : ),  which, 
instead  of  the  sign  of  equality,  may  be  placed  with  one  of  the 
equal  ratios  before  and  the  other  after  it. 

Thus,  4:2  :  :  6:3  expresses  a  proportion. 

Each  ratio  is  a  Couplet,  and  each  term  is  a  Proportional. 

546.  The  Antecedents  of  a  proportion  are  the  antecedents  of 
its  ratios,  and  the  Consequents  are  the  consequents  of  its  ratios. 

547.  The  Extremes  of  a  proportion  are  its  first  and  fourth 
terms,  and  the  Means  are  its  second  and  third  terms. 

When  the  means  of  a  proportion  are  equal,  either  is  called  the  Mean 
Proportional  between  the  extremes. 

548.  A  Missing  Term  in  a  proportion  may  be  denoted  by  x. 

Thus,  in  the  proportion  84  :  8  : :  21  :  a:,  the  missing  term  is  denoted  by 
the  X. 

549.  The  ratio  of  one  number  to  another  is  always  expressed 
by  an  abstract  number ;  hence,  any  couplet  of  a  proportion,  in 
a  process  of  computation,  may  be  regarded  as  formed  of  ab- 
stract numbers. 

550.  Principles. — 1.  The  product  of  the  extremes  is  equal  to 
the  product  of  the  means. 

For,  since  every  ratio  may  be  expressed  in  the  form  of  a  fraction,  the 
proportion  4  :  6  : :  2  :  3  may  be  expressed  thus :  f  =  |.  Keducing  these 
fractions  to  similar  fractions,  we  have  |-^  =|-^'  The  resulting  frac- 
tions being  equal,  and  having  the  same  denominators,  the  numerators 
must  be  equal.  Hence,  4X3=2X6.  But  the  factors  4  and  3  are  tlie 
extremes,  and  tlie  factors  6  and  2  are  the  means.     Hence,  also, 

2.  Either  extreme  is  equal  to  the  product  of  the  means  divided 
by  the  other  extreme;  and 

3.  Either  ynean  is  equal  to  the  produd'of  the  extreme  divided 
by  the  other  mean. 


PROPORTION.  223 

EXEItCISES. 

551.  Find  the  missing  terms  in  the  following  proportions — 


1.  8  :  4  : :  28  :  re  Ans.  14. 

2.  2  :  7  : :  a;  :  10.         Ans.  2f 

3.  9  :  6  : :  6  :  x. 

4.  a:  :  13  : :  35  :  7.       Ans.  65. 

5.  ^7  :  a; : :  9  bu.  :  63  bu. 


6.  3|  :  a;  : :  2  :  f .     Ans.  1^. 

7.  I  :  -J  :  :  f  :  -t.'     '     ^l»i.s.  |. 

8.  3  :  I  : :  I  :  a;.         yins.  yL. 

9.  a;  :  11  T.  : :  S4.32  :  $5.94. 
10.  21  yd.  :  45  yd.  :  :  a;  :  $50. 


SIMPLE  PROPOETION. 

552.  Simple  Proportion  is  the  equality  of  two  simj)le  ratios. 

Simple  proportion  is  sometimes  termed  the  Rule  of  Three,  because, 
when  three  of  its  proportionals  are  given,  they  may  be  used  to  find  the 
fourth. 

553. — Ex.  1.  If  a  man  earn  $24  in  2  mouths,  how  much 
will  he  earn  in  9  months  ? 

$x   :  :  27110.  :9mo.  Solution.  -  There  is  a  ratio 

between  $24  and  %x,  or  required 
Or,  term,  for  they  are  similar,   both 

2  mo.  '.  9  mo.  '.  '.  $24-  •  $x        dollars ;     likewise      between      2 

months   and   9  months,   for  they 
(f^  _  #^  X  P  _  ^  .  ^  n  are  similar. 

money  earned  in  9  months  will  be  as  many  times  the  $24  earned  in  2 
months  as  the  9  months  are  times  2  months. 

Then,  expressing  the  proportion,  we  find  the  unknown  term,  by  Prin. 
3,  Art.  550,  to  be  $108. 

2.  If  6  men  use  a  barrel  of  flour  in  60  days,  how  long  will 
it  last  9  men  ? 

6  men  '.  9  men  :  I  x  days  I  60  days  Soltttion.— If  6   men 

use  a  barrel  of  flour  in  60 
f^  ,  „  days,  9  men  will  use  it  in 

xdays^ -^ =  40  days  f   as    many   as    60    davH. 

3-  Hence,  the   proportion,    6 

m.en  :  9  men  :  :  x  days  :  60  days ;   which  gives  for  the  required  term, 
40  days. 

3.  How  many  casks,  of  32  gallons  each,  will  hold  as  much 
as  48  casks,  of  42  gallons  each  ? 


224  PROPORTION. 

4.  If  18  men  can  mow  a  field  in  12  days,  how  many  men 
can  mow  a  similar  field  in  9  days  ?  Ans.  24. 

554.  Rule  for  Simple  Proportion.— ^S'eZeci^  the  ratios  upon 
which  the  question  depends,  and  foriiv  from  them  a 
proportion. 

TJien,  if  the  required  tenn  he  an  extreme,  divide  the 
prodioct  of  the  T)%eans  hy  the  given  extreme;  or,  if  it 
be  a  mean,  divide  the  product  of  the  extremes  by  the 
given  mean. 

FltOBI^EMS. 

1.  If  a  man  earn  624  in  2  weeks,  how  much  Avill  he  earn  in 
62  weeks?  Ans.  §624. 

2.  When  385  yards  of  muslin  can  be  bought  for  $63,  how 
much  can  be  bought  for  $18  ? 

8.  When  42  casks  of  molasses  cost  $492.66,  what  is  the  cost 
of  5  casks?  Ans.  $58.65. 

4.  If  3  yards  of  cloth  that  is  2\  yards  wide  will  line  a  gar- 
ment, how  much  cloth  that  is  only  f  of  a  yard  wide  will  line 
the  same  garment?  Ans.  10  yards. 

5.  If  17  carpenters  can  do  a  piece  of  work  in  11  jy  days,  in 
what  time  can  7  carpenters  do  it  ? 

6.  If  a  cistern  can  be  filled  in  3  hours  25  minutes  by  2 
pipes,  in  what  time  can  it  be  filled  by  5  pipes  of  like  size  ? 

Ans.  1  hour  22  minutes. 

7.  A  person  travelled  a  certain  distance  in  12  days,  walking 
10  hours  a  day.  How  many  days  would  the  same  journey 
have  required  had  he  walked  only  8  hours  a  day  ?     Ans.  15. 

8.  When  19^  acres  of  land  sell  for  $1800,  for  how  much 
will  9  acres  sell  ? 

9.  How  many  men  would  perform  in  168  days  a  piece  of 
work  which  108  men  can  perform  in  266  days?      Ans.  171. 

10.  When  27  tons  of  hay  cost  $675,  what  is  the  cost  of  21 
tons? 

11.  If  a  ship  have  provisions  sufficient  to  last  24  men 
80  days,  how  long  Avill  the  same  last  if  the  ship  take  on  board 
8  more  men  ?  Ans.  60  days. 


PROPORTION.  225 

12.  If  8  horses  eat  24  bushels  3  pecks  of  oats  in  10  days, 
how  many  bushels  will  6  horses  eat  in  the  same  time  ? 

13.  When  40  yards  of  cloth  cost  $82f,  what  will  ^  of  |  of  a 
yard  cost  ?  Ans.  $.90. 

14.  If  f^  of  a  ship  cost  $42000,  what  will  y^g-  of  a  ship  cost  ? 

15.  When  $7.49  pays  for  ^  of  a  ton  of  coal,  what  will  16| 
tons  cost?  A71S.  $160.50. 

16.  If  the  earth  in  its  orbit  moves  19320  miles  in  16 
minutes  48  seconds,  how  far  does  it  move  in  one  hour  ? 

Ans.  69000  miles. 

COMPOUND   PROPORTION. 

555.  A  Compound  Proportion  is  a  proportion  in  which  one 
or  both  ratios  are  compound. 

rrv,        9:18        „„     1^^  J  12:  3        6:3  , 

Ihus,  „      _    :  :  99  :  165,    and  „      ,  '•  '■  r^    n  are  compound  propor- 
D  :  5  cJ   :  4        9  :  o  v. 

tions. 

556.  The  formation  of  a  compound  proportion  may  often 
be  facilitated,  in  the  solution  of  problems,  by  applying  the 
following 

557.  Principle. — The  ratio  between  two  causes  at  work  for 
the  same  end  will  equal  the  ratio  between  the  effects  produced. 

558. — Ex.  1.  If  2  men  can  mow  15  acres  in  6  days,  how 
many  acres  can  3  men  mow  in  8  days  ? 

2  men  :  3  men       ^^  Solution.— If  2   men 

6  days  :  8  days  ''  ''  ^  ''"''^^  '  ""  '''^^''         can  mow  15  acres,  3  men 

can  mow  f  as  many  acres ; 
15  acres  X*X*  _  ^^  ^^nce,   the    proportion   2 

X  acres  = ^r^'^  —  '^^  acres.  men  :  3  men  :  :  15  acres  : 

*  X  acres. 

If  in  6  days  15  acres  can  be  mown,  in  8  days  there  can  be  mown  |  as 
many  as  15  acres ;  hence,  the  proportion  6  days  :  8  days  : :  15  acres  :  x 
acres. 

By  the  first  proportion,  the  required  term  would  be  |  of  15  acres,  and 
by  the  second  proportion,  |  of  15  acres ;  hence,  by  the  two  combined  it 
will  be  I  of  f  of  15  acres,  or  15  acres  X  I  X  I,  or  30  acres. 


226  PROPORTION. 

Or,  selecting  the  ratios  upon  which  the  question  dei)ends,  and  forming 

2  :  3 
a  proportion,  we  have  g  .  g  •'  =  15  acres  :  x  acres,  or  2  X  6  :  3  X  8  ::  15 

acres  :  x  acres,  which  gives,  as  before,  30  acres  for  the  required  term. 

2.  If  6  men  in  8  weeks  can  build  a  wall  400  feet  long  and  8 
feet  high,  how  long  a  wall  that  is  2  feet  high  can  12  men  build 
in  4  weeks  ? 

Solution. 

12  men     ,  ,  4^0  jeet  long  :  x  feet  long. 
4  weeks  '  '         8  feet  high  :  B  feet  high. 

1st  effect.  ■)         C2d  effect. 

400  \  'A     X 

8)12 

^       ;  ^  400feetXjX^_X4  ^  Applying  the  prin- 

"  -z^  x-0-  x-8-  J  ciple   of    causes  and 

effects  (Art.  557), 
We  have,  as  the  first  cause,  6  men  at  work  8  weeks,  and,  as  the  second 
cause,  12  men  at  work  4  weeks.     These  causes  are  similar,  and  constitute 
a  compound  ratio. 

The  effect  of  the  first  cause  is  a  wall  400  feet  long  and  8  feet  high ;  the 
effect  of  the  second  cause  is  a  wall  x  feet  long  and  2  feet  high.  These 
effects  are  similar,  and  constitute  a  compound  ratio. 

These  compound  ratios  are  equal,  and  form  a  compound  proportion. 
The  required  number  is  a  factor  of  one  of   the    extremes;   hence, 
dividing  the  product  of  the  means  by  the  product  of  the  given  factors  of 
the  extremes,  we  have  that  number,  which  is  1600  feet. 

3.  If  5  compositors,  in  16  days,  11  houio  .  jng,  can  compose 
25  sheets  of  24  pages  in  each  sheet,  44  lines  in  each  page,  and 
40  letters  in  a  line,  in  how  many  days,  each  10  hours  long,  may 
9  compositors  compose  a  volume  to  be  printed  in  the  same 
letter,  consisting  of  86  sheets,  16  pages  to  a  sheet,  50  lines  to  a 
page,  and  45  letters  to  a  line  ? 

SOLtJTION. 

1st  effect 

25 


FBOrORTION.  227 

S 
a;  days ^^xi^x^x^_x.^x^^x^^  ^^  -^ 
s-     -^    ir     s- 

Here,  the  first  cause,  5  compositors  working  16  days  11  hours  long 
:  the  second  cause,  9  compositors  working  x  days  10  hours  long  : :  tiie 
first  effect,  25  sheets  of  24  pages  in  each  sheet,  44  lines  in  each  page,  and 
40  letters  in  a  line  :  the  second  effect,  36  sheets  of  16  pages  in  each  sheet, 
50  lines  in  each  page  and  45  letters  in  a  line. 

Regarding  the  terms  as  abstract,  and  dividing  the  product  of  the  ex- 
tremes by  the  product  of  the  means,  since  the  required  number  belongs 
to  one  of  the  means,  we  have  as  that  number  12. 

559.  Rule  for  Compound  Proportion.—Select  the  ratios  upon 
which  the  question  depends,  and  form  from  them  a 
proportion. 

Tlven,  if  the  required  numher  belongs  to  either  ex- 
treme, divide  the  product  of  the  means  hy  the  product 
of  the  given  extrernes  ;  or,  if  it  belongs  to  either  mean, 
divide  the  product  of  the  extremes  by  the  product  of 
the  given  means. 

PROBLEMS. 

1.  If  you  travel  360  miles  in  12  days  of  8  hours  each,  how 
many  miles  can  you  travel,  at  the  same  rate,  in  60  days  of  6 
hours  each?  Ans.  1350. 

2.  When  12  men  can  harvest  32  acres  of  corn  in  18  days, 
how  many  men,  at  the  same  rate,  can  harvest  128  acres  in  36 
days?  Alls.  24. 

3.  If  2  horses  consume  80  centals  of  corn  in  365  days,  how 
many  centals  will  11  horses  consume  in  73  days,  at  the  same 
rate  ? 

4.  If  6  men  can  build  a  wall  20  feet  long,  6  feet  high  and 
4  feet  thick  in  16  days,  in  what  time  can  24  men  build  one  200 
feet  long,  8  feet  high  and  6  feet  thick  ?  Ans.  80  days. 

5.  If  a  barrel  of  flour  will  feed  a  family  of  8  persons  for  13 
days,  how  many  barrels  will  feed  a  family  of  3  times  as  many 
persons  for  a  year  of  365  days?  Ans.  84^. 


228  PROPORTION. 

6.  When  84  men  can  mow  72  acres  of  grass  in  15  days,  how 
many  acres  can  96  men  mow  in  12  days? 

7.  A  farmer  employed  16  laborers  to  harvest  his  wheat, 
calculating  that  they  would  do  it  in  16  days,  but  after  8  days' 
work  he  resolved  to  have  the  remainder  harvested  in  4  days. 
How  many  additional  laborers  must  be  employed  for  that 
purpose  ? 

8.  If  I  travel  300  miles  in  6  days  of  8  hours  each,  in  how 
many  days  of  10  hours  each  can  I  travel  450  miles,  if  I  travel 
one  half  faster  than  at  first  ?  Ans.  A^  days. 

9.  If  320,000  bricks,  9  inches  long,  5  inches  broad  and  2\ 
inches  thick,  are  required  for  the  construction  of  a  building, 
how  many  bricks,  12  inches  long,  6  inches  broad  and  3  inches 
thick,  would  have  been  required  for  the  same  purpose  ? 

Am.  150,000. 

TEST    QUESTIONS. 

560. — 1.  What  is  Ratio?  How  is  the  ratio  of  one  number  to 
another  ascertained  ? 

2.  What  are  the  Terms  of  a  ratio  ?  What  is  the  first  term  called  ? 
The  second  term  ?     When  is  a  ratio  simple  ?     When  compound  ? 

3.  What  kind  of  numbers  must  the  terms  of  a  ratio  be  ?  Why  may 
both  terms  of  a  ratio  be  either  multiplied  or  divided  by  the  same  num- 
ber without  affecting  its  value  ? 

4.  What  is  Proportion?  What  is  the  sign  of  proportion?  How 
does  it  differ  from  the  sign  of  ratio  ? 

5.  What  are  the  Terms  of  a  proportion  ?  What  is  each  ratio  of  a 
proportion  called  ?     What  is  each  term  of  a  proportion  called  ? 

6.  What  are  the  antecedents?  The  consequents?  The  extremes? 
The  means? 

7.  How  may  a  Missing  Term  of  a  proportion  be  denoted?  How 
may  any  couplet  of  a  proportion  be  regarded  in  computation  ? 

8.  To  what  is  the  product  of  the  means  of  a  proportion  equal  ?  To 
what  is  either  extreme  equal  ?     Either  mean  ? 

9.  What  is  Simple  Proportion?  What  is  it  sometimes  called? 
What  Ls  the  rule  for  simple  projjortion  ? 

10.  What  is  Compound  Proportion?  What  is  the  rule  for  com- 
pound proportion  ? 


DISTRIBUTIVE  PROPORTION.  229 

SECTION    LVII. 
DISTBIB  UTI VE  FR  OF  OR  TIOK. 

561— Ex.  1.  What  number  is  |  of  25  ?    Is  f  of  25  ? 

2.  If  25  apples  be  divided  between  two  boys,  so  that  one 
shall  have  2  as  often  as  the  other  shall  have  3,  what  part  of 
25  apples  will  each  have?     How  many  apples  will  each  have? 

3.  Divide  15  into  two  parts  having  the  ratio  of  2  to  5. 

4.  If  27  cents  be  divided  among  3  boys,  so  that  their  shares 
shall  be  to  one  another  as  2,  3  and  4,  what  part  of  27  cents 
will  the  share  of  each  be  ? 

5.  Two  boys  buy  42  oranges,  and  divide  them  into  two  parts 
in  the  ratio  of  3  :  4.     How  many  wall  there  be  in  each  part  ? 

DEFINITIONS. 

562.  Proportional  Parts  of  a  number  are  such  parts  of  the 
number  as  are  proportional  to  given  numbers. 

Thus,  5,  10  and  15  are  parts  of  30  proportional  to  1,  2  and  3. 

563.  Distributive  Proportion  is  the  process  of  separating  a 
number  into  parts  proportional  to  given  numbers. 

WRITTEN  EXERCISES. 

564. — Ex.  1.  Divide  1035  into  two  parts  which  shall  be  to 

each  other  as  2  to  7. 

ry  ,   ^ Q  Solution. — The  parts  are  to  be  to 

/+^  —  bf  ^^^^  ^^^^^  ^^  2  to  7.     The  sum  of  2 

-Y  1035^=230  ^"^  ^  ^*  ^-     "^^^^  smaller  of  the  parts, 

then,  must  be  |  of  1035,  which  is  230 ; 

-Xl035=  805  and  the  larger  of  the  parts,  |  of  1035, 

^  which  is  805.     Or, 

Or,  Since  the  parts  are  to  be  in  the  ratio 

Q  .   (j)  .  .    1035  '  230       ^^  ^  **'  ^'  ^"^  ^^  sum  of  the  parts  is  9, 
'  ""   ■  '  '  as  9  :  2  : :  1035  :  230,  the  smaller  part; 

9  \  7  '.:  1035  :  805       and,  as  9  :  7  :  :  1035  :  805,  the  larger 
part. 

The  correctness  of   the   solution    is 
230  +  805=^1035  tested  by  adding  the  parts,   for  their 

sum  must  equal  the  given  number. 
20 


280  PARTNERSHIP. 

2.  Divide  500  into  3  parts  proportional  to  the  numbers  2,  3 
and  5. 

565.  Rule  for  Distributive  Proportion.— Ta/ce,  for  the  re- 
quired parts,  such  fractions  of  the  quantity  to  be 
divided  as  each  of  the  numbers  to  which  the  parts  are 
to  be  proportional  is  of  the  suin  of  these  numbers. 

PROBLEMS. 

1.  Two  kinds  of  tea  are  mixed  in  the  ratio  of  15  pounds  of 
Oolong  to  9  pounds  of  Japan,  How  much  of  each  is  there  in 
a  mixture  weighing  120  pounds? 

Ans.  75  lb.  Oolong  and  45  lb.  Japan. 

2.  Gunpowder  contains  f  of  its  weight  of  nitre;  nitre  is 
composed  of  39  parts  of  potassium,  14  of  nitrogen  and  48  of 
oxygen.  How  many  pounds  of  potassium  are  there  in  909 
pounds  of  gunpowder? 

3.  Four  regiments  have  for  duty,  respectively,  60,  90,  150 
and  225  men ;  the  commander  requires  a  special  detail  of  70 
men.     How  many  men  must  each  regiment  furnish  ? 

Ans.  8,  12,  20,  30. 


SECTION  LVIII. 

PARTNERSHIP. 

566.  A  Partnership,  or  a  company,  is  an  association  of  per- 
sons for  the  transaction  of  business. 

567.  Partners  are  the  persons  associated  in  business, 

568.  Capital,  or  Stock,  is  that  which  is  invested  in  the  busi- 
ness. 

569.  A  Dividend  is  that  which  is  divided  among  the  part- 
ners as  their  profits  from  the  business ;  and  an  Assessment  is 
that  required  to  be  paid  by  each  partner  for  increase  of  cap- 
ital, or  for  meeting  expenses  or  losses. 

570.  The  Liabilities  of  a  company  are  its  debts,  and  the 
Assets  are  its  property. 


PARTNERSHIP.  231 

CASE   I. 
Profits  or  Losses  Apportioned  according  to  the  Capital. 

571. — Ex.  1.  A,  B  and  C  entered  into  partnership.  A  put 
in  $1500,  B  $1600,  and  C  $900.  The  company  gained  $500. 
What  was  each  partner's  share  ? 

Solutions. 

$1500  +  $1600  +  $900  =  $4-000  =  entire  capital. 

A^spart  of  the  capital  ^-j^  =g 

4000  ~  5 

C's        "         "  =  -^  =  — 

4.000      40 

A's  share  of  gain  =  ^  of  $500  =  $187.50 

8 

B's        "        "    =^  of  $500  =  $200.00 

5 

C's        "        "    =^of$500  =  $112.50 
40 

Or, 

The  gain  $500  =  ^  =  -  =  12^  c/q  of  the  entire  capital. 
A's  share  of  gain  =  j  of  $1500  -=  $187.50 
B's         «      "     =j  of  $1600  =  $200.00 
C's         "      "     =j  of  $900  =  $112.50 
Or, 

A's  share  of  gain  =  12^%  of  $1500  =  $187.50 

B's         "       "    =12~1o  of  $1600  =  $200.00 
C's        «       «    =12J1o   of  $900  =  $112.50 

Proof  $500M) 

2.  A,  B  and  C  engaged  in  business  together.  A  furnished 
$875,  B  $1680,  and  C  $945.  They  gained  $1000.  What  was 
each  partner's  share  of  the  profits  ? 

Ans.  A's,  $250  ;  B's,  $480 ;  and  C's,  $270. 


232  PAR  TNEBSHIP. 

572.  Rule.  Apportion  the  pj'o fit  or  loss  aTYiong  the  part- 
ner's in  proportion  to  their  capital  in  the  business.    Or, 

Find  such  a  part  of  each  man's  capital  as  the  profit 
or  loss  is  of  the  entire  capital,  and  the  result  is  his 
share  of  the  profit  or  loss. 

PROBLEMS. 

Ex.  1.  Thayer,  Lane  &  Co.'s  profits  are  $6400.  Thayer's 
capital  is  $50000,  and  that  of  each  of  his  two  partners  is 
$15000.     How  much  should  each  receive? 

Am.  Thayer,  $4000,  and  each  of  his  partners,  $1200. 

2.  A  and  B  bought  a  ship  together,  A  contributing  \,  and 
B  the  remainder.  C  was  employed  to  manage  the  ship,  and 
was  to  have  for  his  compensation  half  of  the  profits.  The 
expenses  were  $7500,  and  the  income  $25500.  What  are  their 
respective  shares  of  the  profits  ? 

Ans.  A's,  $2250 ;  B's,  $6750  ;  and  C's,  $9000. 

3.  A  bankrupt,  whose  assets  are  $6800,  owes  M  $4500, 
N  $3600,  and  O  $2100.  How  much  can  he  pay  each  of  these 
creditors?  Ans.  M,  $3000;  N,  $2400;  O,  $1400. 

4.  A,  B,  C  and  D  traded  in  company.  They  furnished  of 
the  capital,  $9500,  $6500,  $4000  and  $5000  respectively.  They 
gained  $6125.     What  was  each  partner's  share  of  the  profits? 

5.  A,  B  and  C  joined  in  a'  speculation  with  a  capital  of 
$1440.  They  gained  $1080,  of  which  A  is  entitled  to  $3  as 
often  as  B  is  to  $5,  and  as  C  is  to  $7.  What  was  the  capital 
and  the  gain  of  each  ? 

Ans.  A's  capital,  $288;  B's,  $480;  C's,  $672. 
A's  gain,  $216;  B's,  $360;  C's,  504. 

CA.SE    II, 

Profits  or  Losses  Apportioned  according  to  Capital  and  Time. 

573. — Ex.  1.  A  goes  into  business  at  the  beginning  of  the 
year  with  a  capital  of  $2000.  After  6  months  he  admits  as  a 
partner  B,  with  a  capital  of  $6000.  If  the  net  profits  at  the 
end  of  the  year  are  $2000,  what  is  each  partner's  share  ? 


PARTNERSHIP.  233 

Solution. 
A's  $2000  for  12  mo.  =  $2^000  for  1  mo. 
B's  $6000  for  6  mo.  =  $36000    "      " 
A'saadE 8  together       ^$60000     "       " 

A 8  share  of  profits  =  ^  =  |  of  $2000  =     $800 

Bs  share  of  profits  ^  ~~  =  §  of  $2000  =  $1200 

Proof        $2000 
Or, 

Interest  of  A's  $2000  for  12  mo.  =  $120 

Interest  of  Es  $6000  for  6  mo.    =  $180 
Interest  of  A^s  and  B's  cajntal         =  $300 

A's  share  of  profiU  ^^^  =  1  of  $2000  =     $800 

B's  share  of  profits  =  J'^|J  =  ^  of  $2000  =  $1200 

Proof        $2000 

2.  Wilson,  Hayes  and  Jones  in  partnership  gained  $936. 
"Wilson  had  in  business  $4680  for  4  months,  Hayes  $5616  for 
5  months,  and  Jones  $2880  for  13  months.  Required  their 
respective  shares  of  the  profits. 

574.  R\i\e.— Apportion  the  profit  or  loss  among  the 
partners  in  proportion  to  the  products  of  their  capi- 
tal by  the  time  in  business.    Or, 

Apportion  the  proUt  or  loss  among  the  partners  Ui 
proportion  to  the  interests  of  their  capital  for  the 
time  in  business. 

PROBLEMS. 

1.  A,  B  and  C  rent  a  pasture  together,  for  which  they  agree 
to  pay  $80.  A  put  in  8  cattle  for  180  days,  B,  6  cattle  for  150 
days,  and  C,  20  cattle  for  123  days.  How  much  of  the  rent 
should  each  pay  ?  Ans.  A,  $24 ;  B,  $15 ;  C,  $41. 

20* 


234  PARTNERSHIP. 

2.  Anson  and  Potter  entered  into  partnership  the  first  of 
January,  and  each  put  in  83000.  The  first  of  April,  Anson 
put  in  $1000;  and  the  first  of  September,  Potter  put  in  8500. 
At  the  end  of  the  year  the  profits  proved  to  be  $2000.  What 
should  be  each  partner's  dividend  ? 

Ans.  Anson's,  $1084.34 ;  Potter's,  $915.66. 

3.  Benson  commenced  business  at  the  beginning  of  the 
year  with  $4500.  April  1,  he  took  into  partnership  Colfax 
with  $5000.  At  the  end  of  the  year  it  Avas  found  necessary  to 
contribute  8660  to  meet  liabilities.  What  should  be  each 
partner's  share  of  the  assessment  ? 

4.  Albert  Whiting,  John  Walker  and  Peter  Woodman  formed 
a  company,  under  the  firm-name  of  Albert  Whiting  &  Co. 
Whiting  put  in  87000  for  10  months ;  Walker,  818000  for  5 
mouths;  and  Woodman  $20000  for  3  months.  They  gained 
813200.     AVhat  was  each  partner's  share  ? 

Am.  Whiting's,  84200 ;  Walker's,  85400  ;  AVood- 
man's,  $3600. 

5.  A  and  B  entered  into  partnership.  At  the  commence- 
ment, A  put  in  85000,  but  at  the  end  of  4  months  took  out 
$3000,  and  continued  the  remainder  in  the  business  6  months 
longer.  At  the  commencement,  B  put  in  $3000,  but  at  the  end 
of  5  months  put  in  84000,  and  continued  the  whole  in  business 
3  months  longer.  On  settlement  it  was  found  that  they  had 
lost  81768.     What  was  each  partner's  share  of  the  loss  ? 

Ans.  A's,  $832,  and  B's,  $936. 


TEST   QUESTIONS. 

675. — 1.  What  is  Distributive  Proportion?  What  are  the  pro- 
portional parts  of  a  number? 

2.  What  is  Partnership?  Who  are  partners?  What  is  the  capital 
or  stock  ? 

3.  What  is  a  Divipkni)  by  a  company?  An  assessment?  What  are 
the  liabilities  of  a  company?     Tlio  assets? 

4.  What  is  the  Rruc  for  apportioning  partnership  profits  or  losses 
when  the  capitals  of  the  partners  are  in  the  business  for  equal  times? 
When  the  capitals  of  the  partners  are  in  the  business  for  unequal  times? 


A  VERA  GE   OF  PA  YMENTS.  •  235 

section  lix. 
'average  of  faymekts. 

576.  Averag'e  of  Payiuent.s  is  the  process  of  fiuding  the  av- 
erage time  for  the  payment  of  several  sums  due  at  diiFerent  times. 

577.  The  Average  Time  is  the  date  on  which  debts  due  at 
different  times  may  be  equitably  discharged  by  one  payment. 

578.  The  Term  of  Credit  is  the  time  whicli  is  to  elapse  before 
a  debt  becomes  due. 

579.  The  Areragre  Term  of  Credit  is  the  time  which  is  to 
elapse  before  the  average  time. 

CASE    I. 

Terms  of  Credit  Bcgiiuiing  at  tlie  Same  Time. 

580.— Ex.  1.  On  January  1,  Belden  Wilder  owes  me  $100, 
due  that  day;  S600,  due  March  1 ;  and  $800,  due  July  1.  At 
what  date  may  he  equitably  cancel  his  indebtedness  ? 

$100  X0  =  $0000  SoLUTioN.-The  $100  being  due 

Pnn  V  f??  -  -      1o>nn  ^^^^'  ^'  '^^^  ^^^^^  of  credit  is  0  months. 

800  X  6'  =     4-800  is  the  same  as  the  credit  of  2  times 

$1500  )$6000  ^^^^'  °^  ^1200,  for  1  month. 

^  ^ The  credit  of  $800  for  G  months  is 

-¥•  the  same  as  the  credit  of  6  times 

-r  -,    ,     ,  ^T       -I         $800,  or  $4800,  for  1  month. 

Jwiuary  1  +  4  mo.  =  Mai)  1  „  ,,  ,.^     »  ^, 

•^  ^  Hence,   the  credit  or  the  entire 

indebtedness,  or  $1500,  is  the  same  as  that  of  $1200  +  $4800,  or  $6000, 

for  1  month,  which  is  equal  to  the  credit  of  $1500  for  as  many  months 

as  $1500  is  contained  times  in  $6000,  or  for  4  mouths.     January  1  +  4 

months  equals  May  1. 

We  may  also  consider  the  debtor  as  entitled  to  the  use,  or  interest,  of 

each  of  the  debts  for  its  term  of  credit.     Hence,  the  following 

Interest  of  $100  for  0  mo.  =$00. 00  Solution.  -  Eeckon- 

"  600     "     Smo.=       6.00  i"g  ^^^^  interest  at   6%, 

onn     u     ^,„.  _    SAOO  t'^e  aggregate  of  interest 

^'^^  ^  ^^^•—    ^^-^^  for  the  terms  of  credit  is 

Debts,  $1500     Total  Int.,  $30.00  $6  +  $24,  or  $30. 


23G  .  AVERAGE    OF  PAYMENTS. 

•  The  interest  of  $1500,  or  the  sum  of  the  debts,  at  the  same  rate  for  1 
month  is  $7.50.  Wilder  should  therefore  have  the  use  of  the  $1500  as 
many  months  from  January  1  as  $7.50  is  contained  times  in  $30,  which 
is  4 ;  and  4  months  from  January  1  is  May  1. 

Any  rate  of  interest  might  have  been  used  in  the  computation,  and 
the  result  would  have  been  the  same. 

2.  Three  debts  are  due  me — one  of  $120  in  5  months,  an- 
otlier  of  $125  in  4  months,  and  a  third  of  $500  in  8  months. 
What  is  the  average  time  of  their  payments  ? 

581.  l{\x\Q%.— Multiply  each  of  the  debts  by  its  term 
of  credit,  and  divide  the  sum  of  the  products  by  the 
sum  of  the  debts ;  the  quotient  will  be  the  average 
term  of  credit.    Or, 

Find  the  iivterest  of  each  debt  for  its  term  of  credit, 
and  divide  the  sum  of  their  interests  by  the  interest 
of  the  sum  of  the  debts  for  one  month  o?^  one  day ; 
the  quotient  will  be  the  average  teimi  of  credit. 

The  date  of  the  debts,  plus  the  average  term  of 
credit,  will  be  the  average  time. 

In  finding  the  average  term  of  credit  when  any  of  the  debts  have 
cent«i,  it  is  customary  to  neglect  them  if  less  than  50 ;  and  if  50  or  more 
to  regard  them  as  $1 . 

In  a  result,  if  there  be  a  fraction  of  a  day,  reject  it  when  less  than  \ ; 
and  when  otherwise,  call  it  1  day. 

TItOJiljEMS. 

1.  A  merchant  owes  $60  due  in  72  days,  $85  due  in  128 
days,  $70  due  in  176  days,  and  $105  due  in  320  days'.  Re- 
quired tlie  average  time  at  which  the  Avholc  will  be  due. 

2.  January  1,  Alfred  Day  bought  bills  of  goods  payable  as 
follows:  $70  at  date,  $110  on  March  2,  $80  on  May  5,  $120 
on  July  20,  $48  on  September  27,  and  $50  on  October  7. 
Recjuired  the  average  time  of  payment.  An&.  May  22. 

3.  July  5,  Johnson  Paterson  bought  bills  of  goods  payable 
as  follows:  $500  on  August  5,  $600  on  Septend)er  5,  and  $1000 
on  September  20.     Required  the  average  tiuie  of  payment. 


AVERAGE    OF   I'A  I'M  EATS.  237 

C^SE   II. 
Terms  of  Credit  Beginning:  .at  Different  Times. 

582. — Ex.  1.  I  bought  goods  of  James  Hunt  &  Co.  as  fol- 
lows :  March  1,  a  bill  of  $500,  ou  4  months ;  March  22,  a 
bill  of  6200,  on  2  months,  and  April  29,  a  bill  of  8680,  on  5 
mouths.    What  is  the  average  time  of  payment  of  the  whole  ? 

Solution. 

• 

3Iarch22+2mo.=May22,  000X0  =$00000 
March  1+4,"  =JuJy  1,  500X^0  =  30000 
April  39 +5  "    =  Sept.  39,      680X130=    88^00 

$1380  )$  108400  (7 8^^ 

9660 
11800 
May  33  +  79  days  =  Aug.  9.  llOJfi 


760 


Interest  of  $300  for  0  days      = 

$00.00 

"           500    "   40  days    = 

3.33 

«           680    "   130  days  = 

14.73 

Or, 


Sum  of  bills,  $1380         Total  interest,  $18.06 

Interest  of  $1380  for  1  day  =  $.33 
$18.06  -^  $.33  =  78§.      May  33  +  79  da.  =Aug.  9. 

The  several  bills  are  due  July  1,  May  22  and  September  29,  respectively. 

Selecting  the  earliest  day  of  maturity  as  the  day  from  which  to  reckon, 
$200  has  no  term  of  credit,  the  $500  has  a  credit  of  40  days,  and  the  $680 
has  a  credit  of  130  days,  from  May  22.  The  average  term  of  credits,  by 
either  form  of  solution,  is  79  days,  nearly. 

Hence,  May  22  -f-  79  days,  which  is  August  9,  is  the  average  time 
required. 

The  first  form  of  solution  is  called  the  Product  Method,  and  the  second 
the  Interest  Method.     Accountants  generally  prefer  the  latter. 

The  date  of  the  first  debt's  maturity  was  selected  to  reckon  from  for  con- 
venience. Had  the  latest  date  of  maturity  been  selected,  the  average 
time  would  have  been  counted  back  from  that  date. 


238 


AVERAGE    OF   PAYMENTS. 


2.  Kobert  Hendricks  gave  me,  June  4,  a  note  for  $315.63  on 
4  months ;  June  15,  a  note  for  $535.47  on  2  months ;  'and 
July  3,  a  note  for  $300  on  3  months.  Regarding  these  notes 
without  grace,  should  he  wish  to  take  them  up  by  giving  one 
note  for  their  amount,  when  should  it  be  payable  ? 

Ans.  September  10. 

583.  ^w\b.—Find  the  date  at  which  each  debt  becomes 
due.  Select  the  earliest  date  (it  which  any  of  the  debts 
matures,  and  rechoning  fj^om  it,  as  in  the  previous 
case,  find  the  average  term  of  credit;  and  the  selected 
date,  plus  the  average  term  of  credit,  will  be  the 
average  time. 

In  working  by  the  interest  method,  it  may  be  most  convenient  to  take 
for  the  selected  date  the  ^rst  day  of  the  month  in  which  the  first  credit 
begins. 

When  the  terms  of  credit  are  all  equal,  we  may  simply  find  the  aver- 
age date  of  the  debts,  and  add  the  common  term  credit,  for  the  average 
time. 

jphoblems 

1.  When  should  a  note  to  settle  the  following  account  be 
made  payable  ? 

Lewis  Manly, 

To  Stone,  Dexter  &  Co,     Dr. 


1871. 

May 

13 

To  Merchandise  @  .^  mo., 

as  per  bill 

<$soo 

15 

<( 

u 

"            "           @  2  mo., 

« 

800 

00 

June 

15 

Cash 

99 

83 

$899 

98 

Ans.  August  17. 

2.  Purchased  of  Jonas  Mungor,  on  a  credit  of  90  days, 
January  6,  a  bill  of  $G00,  and  February  15,.  a  bill  of  $200. 
Required  the  average  date  of  purchase  and  the  average  time 
of  payment.  Ans.  January  16 ;  April  16. 


AVERAGE   OF  PAYMENTS. 


239 


3.  What  is  the  average  time  of  the  following  bills,  allow- 
ing to  each  term  of  credit  3  days'  grace  ? — Sold,  April  3,  a  bill 
of  $500  on  3  months ;  April  4,  a  bill  of  $200  on  2  months ; 
April  4,  a  bill  of  |200  for  cash ;  and  April  10,  a  bill  of  $500 
on  3  months.  Ans.  June  21. 

CASE   III. 
Debit  and  Credit  Accomit. 

584. — Ex.  1,  From  what  time  should  a  note  draw  interest 
for  the  balance  of  the  following  account,  allowing  3  days'  time 
to  the  i^ms  on  time  ? 

Dr.    James  Blake  in  account  with  George  Hill.     Cr. 


1870. 

1870. 

May 

8 

To  Mdse.,  Cash 

$  40 

00 

May 

10 

By  Cash 

$  30 

00 

" 

17 

"         "  30day.<i 

240 

00 

" 

80 

it        a 

205 

00 

« 

20 

"    "  Cash 

240 

00 

June 

13 

11        ti 

165 

00 

Solutions. 


Int.  of  $40  for  0  days  =  $0.00 
"  240  "  12  days  =  .48 
"        240    "    42  days  =    1.68 


l7it.  of    $30  for    2  days  =   $.01 
"  205  "     22  days  =      .7516 

"  165  "    36  days  =      .99 


$520                           $2.16 
400                             1.7516 

$400 

Int.  of  $120 for  Ida. 
May  8  ^20  da.  =  May  28. 

May  10,    $30  X  2  - 

"    30,    205X22-- 

June  13,    165  X  36  = 

$400 
$2450 -^  $120  =  20^\ 

$1.7516 

Balance,  $120                         $  .4O84 

$.4084-^.02  =  20.42.        I 
Or, 

May   8,    $40  x    0  =  $00000 

"      20,    240  X  13=      2880 

June  19,    240x4^=    10080 

=  $.02 

=  $60 
=  4510 
=  5940 

$520               $12960 
400                 10510 

Balance,  $120                $2450 

$10510 

May  8  +  20  da.  =  May  28,  the  average  time. 

The  item  on  the  debit  side,  on  30  days,  by  adding  3  days'  grace,  is  due 
June  19.     The  other  items,  being  cash,  are  due  at  their  respective  dates. 

Selecting  the  earliest  date  of  maturity  of  any  item  as  that  from  which 
to  reckon,  we  find  that  the  balance  of  account,  $120,  and  the  balance  of 
interest,  $.408]^,  are  on  the  debit  side.     Were,  then,  Blake  to  settle  at 


240 


AVEEAGB   OF  PAYMENTS. 


that  date  by  paying  the  balance,  he  evidently  would  lose  the  use  of  $120 
for  as  many  days  as  would  be  equivalent  to  $.408y%  interest.  Hence,  the 
balance  is  not  equitably  due,  or  subject  to  interest,  till  as  many  days  after 
May  8  as  would  allow  $120  to  earn  $.408y*o  interest.  This  we  find  to  be 
20  days,  and  May  8  +  20  days  gives  May  28  as  the  required  time. 

Had  the  balance  of  account  and  of  interest  been  on  opposite  sides  of 
the  account,  then  the  balance  would  have  been  due,  or  subject  to  interest, 
earlier  than  that  date  as  many  days  as  would  aUow  the  balance  of  account 
to  earn  the  balance  of  interest. 

Or,  reckoning  from  the  earliest  date  of  maturity  of  any  of  the  items, 
we  find  that  the  balance  of  account,  which  is  $120,  and  of  credit,  which 
is  that  of  $2450  for  1  day,  are  both  on  the  debit  side.  Hence,  Uie  debtor, 
by  paying  $120  May  8,  would  lose  the  credit  of  $2450  for  1  day,  which 
is  the  same  as  the  credit  of  $120  for  as  many  days  as  $120  is  contained 
times  in  $2450,  or  for  20  days. 

Hence,  the  average  time  of  payment  is  May  8  +  20  days,  or  May  28. 

585.  Rule. — Select  the  earliest  date  at  ivhieh  any  of 
the  items  mature,  and,  rechoning  from  it,  find  the  in- 
terest of  each  item  from  that  date  to  its  maturity. 

Divide  the  balance  of  interest  by  the  interest  of  the 
balance  of  items  for  1  day,  and  the  quotient  ivill  be 
the  average  term  of  credit,  which  must  be  counted 
forward,  frorn  the  selected  date,  for  the  average  time, 
when  the  balance  of  interest  is  on  the  larger  side  of 
the  account,  but  backward  when  the  balance  of  inter- 
est is  on  the  smaller  side. 

PROBLEMS. 

1.  When  will  the  balance  of  the  following  account  average 
due,  allowing  3  clays'  grace  on  each  merchandise  item  ? 


Dr. 


J.  B.  Halstead. 


Or. 


1870. 

1870. 

Mar. 

10 

To  Mdse. 

on  30  days. 

$100 

00 

Mar. 

U 

By  Mdse.,  on  SO  days. 

$soo 

00 

April 

15 

20 

"  Oath 

100 
100 

00 
00 

13 

150 

00 

Ans.  March  3,  1870. 


AVERAGE   OF  PAYMENTS. 


241 


2.  What  should  be  the  date  of  a  note  drawing  interest  for 
the  balance  of  the  following  account,  allowing  3  days'  grace  on 
each  item  ? 


Dr. 


R.  B.  Alfred. 


Or. 


1870. 

1870. 

May 

1 

To  Mdse.,  on  30  days, 

$m 

00 

May 

SI 

By  Mdse.,  on  30  days. 

$180 

00 

" 

IS 

"        " 

100. 

00 

" 

'• 

220 

00 

" 

30 

" 

100 

00 

" 

" 

"          " 

50 

00 

Ans.  July  7,  1870. 


CASH  BALANCE  OF  ACCOUNTS. 

586.  The  Balance  of  an  Account,  in  equity,  is  entitled  to  in- 
terest from  the  time  of  its  becoming  due.     Hence, 

If  settlement  of  an  account  is  made  after  the  average  time, 
the  Cash  Balance  is  the  balance  of  items  plus  the  interest  from 
the  average  time  to  the  time  of  settlement. 

If  settlement  is  before  the  average  time,  the  Cash  Balance  is 
the  balance  of  items  minus  the  interest  from  time  of  settle- 
ment to  the  average  time. 

X:XERCISES. 

587. — Ex.  1.  The  balance  of  items  of  an  account,  which  is 
$50,  is  due  by  average,  March  3,  1870.  What  should  be  its 
cash  value,  April  20,  interest  at  6%  ?  Ans.  $50.40. 

2.  The  balance  of  items  of  an  account,  which  is  $130,  is 
due  by  average,  June  17,  1871'.  What  was  the  cash  balance 
June  2,  interest  at  6%  ?  Ans.  S129.67|. 

3.  What  is  the  cash  balance  of  the  following  account, 
Oct.  16,  at  7%? 


Dr. 


Hall,  Weston  &  Co. 


Or. 


1871. 

1871. 

May 

5 

To  Mdse.,  on  3  mn.. 

$19 

83 

June 

20 

By  Mdse.,  on  h  mo.. 

$25 

00 

July 

15 

"        "      ow  /J  mo.. 

UO 

00 

Aug. 

10 

"        "      (m  3  mo.. 

100 

00 

Sept. 

25 

"        "      cm  2  mo.. 

60 

17 

Oct. 

r 

"        "      on  2  mo., 

120 

00 

242  AVERAGE  OF  PAYMENTS. 

TEST    QUESTIONS. 

588.— 1.  What  is  Average  of  Payments?  What  is  the  average 
time  ?     The  term  of  credit  ?     The  average  term  of  credit  ? 

2.  What  are  the  Rules  for  finding  the  average  term  of  credit  when 
the  t«rms  of  credit  begin  at  the  same  time  ?  What  will  be  the  average 
time? 

3.  When  the  terms  of  credit  begin  at  different  times,  how  is  the  aver- 
age term  found  by  the  product  method  ?  By  the  interest  method  ? 
What  is  the  rule  for  finding  the  average  time? 

4.  What  method  is  generally  preferred  by  accountants  in  finding  the 
average  time  of  payment  of  a  debit  and  credit  account?  Why?  What 
is  the  rule  ? 

5.  To  what  is  the  Balance  of  an  Account  entitled  in  equity  from 
the  time  of  becoming  due  ?  What  is  the  cash  balance  of  an  account 
when  the  settlement  is  made  after  the  average  time  ?  When  the  settle- 
ment is  made  before  the  average  time? 


SECTION    LX. 

STOCKS  AJ^D  IXYESTMEMTS. 

589.  Stock  is  money  or  property  employed  in  business,  or 
invested  in  a  company  or  in  a  public  debt. 

590.  The  Par  Yalue  of  stock  is  its  face  or  nominal  value, 

591.  The  Market  Value  of  stock  is  the  sum  it  will  bring 
when  sold. 

592.  A  stock  is  at  par  when  it  sells  for  its  original  or  face 
value,  or  100%  ;  above  par,  or  at  a  premium,  when  it  sells  for 
more  than  its  face  value,  or  above  100%  ;  and  below  par,  or  at 
a  discount,  when  it  sells  for  less  than  its  face  value,  or  less  than 
100%. 

Thus,  when  stock  is  at  par,  it  is  quoted  at  100  ;  when  at  dfc  above  par, 
at  101) ;  and  when  at  6%  below  par,  at  94. 

593.  When  ])aper  money  is  depreciated,  fJold  ceases  to  be  a 
circulating  medium,  and,  like  stocks,  becomes  an  object  of 
investment. 


STOCKS  AND   INVESTMENTS.  243 

Thus,  when  gold  is  quoted  at  109,  $1  of  gold  is  of  the  same  value  a.s 
$1.09  of  currency. 

594.  A  Broker  is  a  dealer  in  stocks,  bonds,  gold,  etc. 

The  usual  rate  of  brokerage  for  buying  or  selling  is  from  xV%  to  |% 
of  the  par  value. 

COKPORATE    STOCKS. 

595.  A  Corporation  is  a  company,  or  an  association  of  per- 
sons, authorized  by  law  to  transact  business  jointly. 

596.  A  Corporate  Stock  is  the  property  invested  in  a  corpo- 
ration. 

597.  A  Share  is  one  of  the  equal  parts  into  which  a  corpo- 
rate stock  is  divided. 

The  Par  Value  of  a  share  is  usually  $100. 

598.  An  Installment  is  a  certain  part  of  the  stock  of  an 
incorporated  company  paid  at  a  particular  time. 

599.  An  Assessment  is  a  sum  which  stockholders  are  called 
upon  to  pay,  on  each  share  held,  to  meet  losses  or  to  make  up 
deficiencies. 

600.  The  Gross  Earnings  of  a  company  are  its  entire 
receipts ;  the  Net  Earnings  of  a  company  are  what  remains 
after  deducting  expenses ;  and  the  Dividend  is  the  sum  paid  to 
the  stockholders  from  the  profits  of  the  business. 

601.  Dividends  and  assessments  are  usually  reckoned  at  a 
certain  per  cent,  of  the  par  value. 

GOVERNMENT    SECURITIES. 

602.  Bonds  are  obligations  securing  the  payment  of  a  cer- 
tain sum  of  money  on  or  before  a  specified  time. 

These,  when  issued  by  Government  or  corporations,  bear  interest  pay- 
able at  fixed  dates. 

603.  Coupons  are  interest  certificates  attached  to  bonds. 
They  are  to  be  cut  off  and  presented  for  payment  when  the 
interest  is  due. 


244  STOCKS  AND    INVESTMENTS. 

604.  Treasury  Notes  are  notes  issued  by  the  Government, 
payable  on  demand,  without  interest,  or  payable  at  a  specified 
time,  with  interest. 

605.  United  States  Goyernment  Securities  consist  of  bonds 
and  Treasury  notes. 

6's  of  1881  are  bonds  which  are  payable  in  1881.  The  interest  on 
them  is  at  the  rate  of  6  %  in  gold,  and  is  payable  semi-annually. 

5-20's  are  bonds  which  are  payable  after  20  years,  and  are  redeemable 
after  5  years,  from  their  issue.  The  interest  on  them  is  at  the  rate  of  6% 
in  gold,  and  is  payable  semi-annually. 

10-40's  are  bonds  which  are  payable  after  40  years,  and  redeemable 
after  10  years,  from  their  issue.  The  interest  on  them  is  at  the  rate  of 
5%  in  gold,  and  is  payable  semi-annually  on  $500  and  $1000  coupon 
bonds,  and  annually  on  registered  bonds,  and  on  $100  and  $50  coupon 
bonds. 

5's  of  1881  are  bonds  which  are  payable  after  1881.  The  interest  on 
them  is  at  the  rate  of  f>fc  in  gold,  and  is  payable  quarterly. 

4|'s  of  1886  are  bonds  which  are  payable  after  1886.  The  interest  on 
them  is  at  the  rate  of  4J  %  in  gold,  and  is  payable  quarterly. 

4's  of  1901  are  bonds  which  are  payable  after  1901.  The  interest  on 
them  is  at  the  rate  of  4%  in  gold,  and  is  payable  quarterly. 

606.  Bonds  issued  by  cities,  counties,  States  and  corpora- 
tions are  usually  named  according  to  the  rate  of  interest  they 
bear. 

Thus,  Virginia  6's  are  bonds  bearing  interest  at  6fo,  issued  by  the 
State  of  Virginia. 

WBITTEN   EXERCISES. 

607. — Ex.  1.  What  is  the  cost,  including  brokerage,  of  400 
shares  of  railroad  stock,  at  95%  ? 

S0I>UTI0N. 

(95%  +  j%)  of  $100  -  $95.25,  cost  of  1  share. 

$95.25  X  400  -  $38100,  cost  of  4OO  shares. 

2.  What  is  the  market  value  of  50  shares  of  National  Bank 
stock,  at  115? 


STOCKS  AND  INVESTMENTS.  245 

3.  How  much,  including  brokerage,  must  be  paid  for  $1000 
Maine  6's  at  101  ? 

Solution. 

(101^0+ jfc)  of$l  =  $1.0lj,  amount  paid  for  $1. 
$1.01JX1000=$1012.50,  "  «      $1000. 

4.  When  gold  is  quoted  at  11 2^,  what  is  the  value  in  cur- 
rency of  $5000  in  gold  ?  Ans.  $5625. 

5.  I  bought  Government  securities  of  the  par  value  of  $3000 
at  lOOf ,  and  sold  them  at  109^.     How  much  did  I  gain  ? 

Am.  $255. 

6.  When  gold  is  worth  112^,  what  is  the  value  in  gold  of 
$5625  in  currency  ? 

Solution. 
At  112^<fc,  $1-1'2^  in  currency  =  $1  in  gold. 
$5625  -^  $1.12^  =  5000,  the  number  of  dollars  in  gold. 

7.  When  gold  is  at  150,  what  is  the  value  in  gold  of  $1  in 
currency?  Ans.  $.66|. 

8.  When  gold  is  at  125,  what  is  the  value  in  currency  of 
$1000  in  gold  ? 

9.  When  the  cost  of  400  shares  of  railroad  stock,  including 
brokerage,  is  $38100,  what  is  the  market  value  per  share  ? 

Solution. 
-%  of$100X400  =  $100,  the  brokerage. 
$38100  —  $100  =  $38000,  market  value  of  4OO  shares. 
$38000  -^  400  =  $95,  market  value  of  1  share. 

10.  For  how  many   shares  of  telegraph   stock,   including 
brokerage,  will  $1524  pay  when  the  stock  is  selling  at  95? 

Ans.  16. 

11.  For  how  many  shares  of  telegraph  stock,  including  the 
brokerage,  will  $1524  pay,  the  stock  being  at  5%  discount? 

21* 


246  STOCKS  AND  INVESTMENTS. 

12.  I  bought  120  shares  of  stock  at  98,  and  paid  the  broker- 
age.    How  much  did  I  gain  by  selling  the  same  at  103  ? 

13.  I  bought  50  shares  of  stock  at  95,  and  sold  them  at  par, 
receiving  in  the  mean  time  a  dividend  of  4%.  What  was  the 
profit?  Am.  S450. 

14.  When  a  certain  5%  stock  is  at  85,  how  much  must  be 
invested  in  it  to  yield  an  annual  income  of  $650  ? 

Solution. 
$100Y.  .05  =  $5,  income  of  1  share  of  5%  stock. 
$650  -^  $5  =  130,  number  of  shares  to  yield  $650. 
$85  X  130  =  $11050,  the  sum  which  must  be  invested. 

15.  What  sum  must  be  invested  in  U.  S.  4^%  bonds,  at  96, 
brokerage  at  ^%,  to  secure  an  annual  income  of  $900? 

Ans.  $19250. 

16.  When  U.  S.  10-40's  are  quoted  at  106,  what  sum  must 
be  invested  in  them  to  secure  an  annual  income  of  $400  ? 

17.  How  much  must  be  invested  in  Central  Railroad  7's,  at 
90,  to  produce  an  annual  income  of  $350  ?  Ans.  $4500. 

18.  What  per  cent,  on  the  investment  will  an  8%  stock,  at 

120,  yield? 

Solution. 

$1  of  stock  at  100  yields  8%. 

$1  at  120  must  yield  j^  of  8%,  or  6^%. 

19.  What  per  cent,  will  U.  S.  4's,  purchased  at  80,  yield  ? 

20.  What  is  the  rate  of  income  upon  money  invested  in 
railroad  7's,  at  87^  ? 

21.  Which  will  pay  the  better  rate  of  income — ^U.  S.  4^'s,  at 
89f,  or  a  6%  stock,  at  119f ,  the  brokerage  of  each  being  \</o  ? 

Solution. 

Income  of  4^ - ^-f^-  or  f^  of  4\lo  =5%. 

*     i 

Ana.  Income  same  from  both. 


EXCHANGE.  247 

22.  At  what  rate  must  I  buy  a  5%  stock  that  I  may  receive 
8%  on  my  investment? 

23.  For  how  much  above  par  must  a  6%  stock  sell  to  pay 
an  interest  of  5%  on  the  investment?  Ans.  20%. 

24.  For  how  much  below  par  must  a  7  %  stock  be  bought 
to  pay  an  interest  of  8%  on  the  investment? 

25.  At  what  rate  must  U.  S.  5's  be  bought  to  secure  4^-% 
on  the  investment  ? 

26.  What  rate  of  interest  does  an  investment  in  U.  S.  4i's, 
at  96,  pay  ? 

27.  When  gold  is  at  120,  what  must  be  the  price  of  U.  S. 
5-20's  to  yield  b^%  in  currency  ?  Ans.  130}^. 


SECTION    LXI. 
EXCEAJ^GE. 

608.  A  Draft  is  a  written  order  by  one  party  to  another  to 
pay  a  certain  sum  of  money  to  a  third  party  or  to  his  order. 

609.  The  Drawer  is  the  maker  of  the  draft. 

610.  The  Drawee  is  the  party  ordered  to  pay. 

611.  The  Payee  is  the  party  to  whom,  or  to  whose  order,  the 
draft  is  payable. 

The  drawee  accepts  a  draft  by  writing  his  name  across  the  face  of  it. 
This  denotes  his  agreement  to  pay  the  draft. 

The  payee  may  assign  a  draft  by  writing  his  name  upon  the  back  of 
it.     This  makes  him  security  for  its  payment. 

612.  A  Sight  Draft  is  a  draft  payable  when  presented. 

613.  A  Time  Draft  is  a  draft  payable  at  a  time  named  in  the 
draft. 

Time  drafts  are  entitled  to  three  days  of  grace,  but  grace  is  not  usually 
allowed  on  sight  drafts. 

When  a  draft  is  drawn  at  usance,  it  is  entitled  to  the  time  allowed  by 
custom  or  by  the  law  of  the  place  where  it  is  payable. 

614.  Exchange  is  the  process  of  remitting  value  from  one 
place  to  another  by  means  of  drafts. 


248  EXCHANGE. 

FOEM  OF  A   DKAFT. 


75  ^  Wilmington,  Aug.  4,  1871. 

100 

Ten  days  after  Sight,  pay  to  the 


order  of  Mooi^e,  Merrill  ^  Co., 

Five  Hundred  Sixty -Five —Dollars. 

Value  received,  and  charge  to  account  of 

Henry  Hartwell. 
To  Cooh  ^'  Farnham, 

Washin^toit. 


Omit  in  the  above  the  words  "  ten  days  after  sight,"  or  insert  in  their 
stead  "  at  sight,"  and  the  form  is  that  of  a  Sight  Draft. 

615.  The  Course  of  Exchange  is  the  variation  between  the 
face  of  a  draft,  or  bill,  and  its  cost. 

Exchange  is  at  par  when  a  draft  sells  for  its  face  ;  at  a  premium  when 
it  sells  for  more  than  its  face ;  and  at  a  discount  wlien  it  sells  for  less 
than  its  face. 

DOIklESTIC    OR    INLAND    EXCHANGE. 

616.  Domestic  or  Inlaud*Bills  are  drafts  payable  in  the 
country  in  Avhich  they  are  draAvn. 

617.  In  the  Computation  of  the  cost  of  an  inland  bill,  the 
Base  is  the  face  of  the  bill,  and  the  Bate  is  the  rate  per  cent, 
of  premium  or  discount. 

The  cost  of  a  bill  at  par  is  the  face  of  the  bill ;  at  a  pre- 
mium, the  face  plus  the  premium  ;  and  at  a  discount,  the  face 
minus  the  discount.     Hence  the  following — 

618.  Rules  for  Inland  Exchange.— i.  To  find  the  cost  of  a 
hill,  multiply  the  cost  of  >$!  of  exchange  by  the  num- 
ber denoting  the  face  of  the  bill. 

2.  To  find  the  face  of  a  bill  that  can  be  bought  for 
a  given  sum,  divide  the  given  sum  by  the  ivumber 
denoting  the  cost  of  $1  of  e.vchange. 


EXCHANGE.  249 

PROBLEMS. 

1.  What  will  be  the  cost  of  a  sight  draft  on  New  York  for 
$5850,  at  1%  premium  ?  Am.  $5879.25. 

2.  When  exchange  is  at  ^%  premium,  what  is  the  face  of  a 
sight  draft  that  costs  $5879.25  ? 

3.  How  much  must  be  paid  in  Pottsville  for  a  draft  of  $750 
on  Pittsburg,  exchange  being  at  l|-%  discount? 

Ans.  $738.75. 

4.  What  is  the  face  of  a  draft  that  can  be  purchased  for 
$4301,  when  exchange  is  at  2^%  discount?         Ans.  $4400. 

5.  What  must  be  paid  in  Vicksburg  for  a  draft  of  $600  on 
St.  Louis,  at  60  days,  exchange  being  at  101,  and  interest  at 
9%? 

Solution. 

Bank  discount  of  $1  for  63  days,  at  9%  =  $.01575. 

$1  -  $.01575  =  $.98435,  cost  of  $1  at  par. 
$.98425  +  $.01  =  $.99425,  cost  of  $1  at  101. 
$.99425  X  600  =  $596.55,  cost  of  the  draft. 
Or, 

Bank  discount  of  $600  for  63  days,  at  9%  =  $9.45. 
$600-$9.45=-$590.55,  cost  of  draft  at  par. 
$600  X  .<9i  =  $6,  premium  of  draft  at  101. 
$590.55  +  $6  =  $596.55,  cost  of  the  draft. 

6.  What  will  be  the  face  of  a  draft  payable  60  days  after 
sight,  that  can  be  bought  for  $596.55,  exchange  being  1  %  pre- 
mium, and  interest  9  %  ? 

7.  How  much  must  be  paid  for  a  draft  of  $750,  payable  10 
days  after  sight,  exchange  being  at  |%  discount,  and  interest 
iit  6%  ?  Ans.  $744,621. 

8.  I  owe  a  note,  in  New  York,  of  $3000,  with  interest  for  1 
year  at  7%.  What  must  be  the  face  of  a  sight  draft,  exchange 
at  101^,  which  I  can  remit,  and  thus  exactly  discharge  the 
note  and  interest  ?  J.m.  $3250.12^. 


250  EXCHANGE. 

FOREIGN    EXCHANGE. 

619.  Forei^  Bills  are  drafts  drawn  in  one  country  and  pay- 
able in  another. 

620.  The  Par  of  Exchange  between  two  countries  is  the  vahie 
of  the  currency  of  one.  country  estimated  in  the  currency  of 
the  other. 

621.  Bills  of  Exchange  between  the  United  States  and  foreign 
countries  for  convenience  are  generally  drawn  and  negotiated 
on  London  or  Paris. 

Foreign  bills  are  drawn  in  sets  of  three,  of  the  same  tenor  and  date, 
and  called,  respectively,  First,  Second  and  Third  of  Exchange.  They  are 
forwarded  diflerently,  to  prevent  delay  by  accident.  When  one  is  paid, 
the  others  are  cancelled. 

BILLS  ON  ENGLAND. 

622.  Bills  on  England,  or  London,  are  drawn  in  English,  or 
sterling  money,  the  denominations  of  which  are  shown  in 
the  following — 

TABLE. 

4-  farthings  (qr.or  far.)  are  1  penny  .  .  .  d. 

12  pence  "    1  shilling. .  s. 

2  shillings  "    1  florin  .  .  .  fl. 

10  florins,  o?'  20  shillings, "    1  pound .  .  .  £. 

623.  The  Value  of  a  pound  sterling,  which  is  represented  by 
the  English  gold  coin  called  the  Sovereign,  previous  to  the 
change  in  the  United  States  coinage  in  1834,  was  $4f,  or 
$4.44A 

In  the  present  gold  coinage  of  the  United  States  a  sovereign 
of  standard  weight  is  equal  to  $4.8634.  Allowing  for  the  wear 
of  coins,  we  have  what  the  Government  has  established  as  the 
Custom-house  or  legal  value  of  the  pound,  which  is  $4.84. 

624.  In  the  Computation  of  Exchange  the  old  value  of 
$4^  is  usually  considered  the  Base. 

Hence,  when  English  excliange  is  quoted  at  109,  it  is  at  the  Custom- 
house value,  and  when  at  109|,  it  is  at  about  the  intrinsic  value,  or  true 
par. 


EXCHANGE.  251 

WRITTEN  EXERCISES. 

625.— Ex.  1.  What  is  the  cost  in  New  York  of  a  bill  of 
exchange  on  London  of  £400  10  s.  6  d.,  at  109f,  including 
brokerage  at  :^%  ? 

Solution.  —  £400  10  s. 
£400  10  s.  6  d.  =  £400.525  ed.,  decimally  expressed,  is 

*//,  V  7  M  ^  £400.525.   £1,  at  109|,  and 

^^^^X400.525  =  $1958.12+      ^.^kerage  at  \7o,  will  cost 

$40X1.10. 


Hence,  £400.525  will  cost  400.525  times  ^^"V^"'  o'^  $1958.12  +. 

2.  What  is  the  face  of  a  bill  on  London  which,  at  109f ,  and 
brokerage  at  \%,  can  be  bought  for  $1958.12? 

<t^1Q^S  10?  .  $40^  1-10  _  /nn  p:9^  Solution.— £1  pf  face  of 

ipitfDiS.l^-^         g         -.^UU.O^O        g^^,^^nge^   at   9|   premium, 

£400.525  =  £400  10  S.  6  d.        can  be  bought  for  ?iO-=^  J-i». 

Hence,  as  many  pounds 
of  face  can  be  bought  for  $1958.12  as  is  denoted  by  the  quotient  of 
1958.12  H-  ?±0J<i-iQ,  or  £400.525  =  £400  10  s.  6  d. 

3.  What  will  a  bill  on  London  for  £2000  cost,  at  108  ? 

626.  Rules  for  Exchange  on  England.— i.  To  find  the  cost 
of  a  hill,  multiphj  the  cost  of  £1  at  the  given  rate  by 
the  number  denoting  the  face  of  the  bill,  expressed  in 
pounds  and  the  decimal  of  a  pound. 

2.  To  find  the  face  of  a  bill  that  can  be  bought  for  a 
given  sum,  divide  the  given  sum  by  the  cost  of  £1  of 
exchange  at  the  given  rate. 

FMOBIEMS. 

1.  What  will  be  the  cost  in  United  States  money  of  a  bill 
on  England  of  £7000,  exchange  at  111  ?    Am.  $34533.33i. 

2.  When  exchange  is  at  108^,  how  much  English  money 
will  $4822|  purchase  ?  Ans.  £1000. 

3.  When  exchange  is  at  111,  what  will  be  the  face  of  a  bill 
on  England  that  can  be  purchased  for  $34533.33^  ? 


252 


EXCHANGE. 


4.  How  much  must  be  paid  in  United  States  currency  for  a 
sterling  bill  of  £505  15  s.  6  d.,  exchange  at  108|,  brokerage  \%, 
and  gold  at  110? 

BILLS  ON  FKANCE. 

627.  Bills  on  France,  or  Paris,  are  drawn  in  the  money  of 
France,  the  denominations  of  which  are  as  in  the  following — 

TABLE. 
100  centimes  are  1  franc  .  .  .  fr. 

628.  Francs  and  centimes  may  be  written  together  after  the 
manner  of  dollars  and  cents. 

Thus,  25.20  fr.  expresses  twenty-five  francs  twenty  centimes. 

629.  The  Franc  in  silver,  which  is  the  standard  in  France, 
is  valued  at  the  United  States  Custom-house  at  $.186. 

The  intrinsic  value  of  a  franc  in  silver,  if  new,  is  $.196,  so 
that  about  b.lO\  francs  are  equal  to  $1. 

630.  Exchange  on  France  is  quoted  at  a  certain  number  of 
francs  to  a  dollar. 

Thus,  when  exchange  on  Paris  is  quoted  at  5.16,  it  is  at  5.16  francs  to 
a  dollar,  and  is  above  par. 

The  value  in  the  United  States  of  some  of  the  Foreign  Coins 

often  named  in  quotations  of  foreign  markets  is  given  in  the 

following — 

TABLE. 


Places. 


Great  Britain 
France  . 
Prussia  . 
Germany 
Bussia  . 
Portugal 
Spain  . 
Turkey  . 
Mexico  . 
East  Indies 
China     .     . 


Name  of  Coin. 


I  pound  =  20  shillings  . 
1  franc  =  100  centimes 
1  thaler  (new) .  .  .  , 
1  florin  or  guilder     .     . 

1  rouble 

1  milrei 

1  real     

1  piastre 

J  dollar  {7ierv)       .     . 

1  rupee 

1  tael , 


Value. 


.196 
.7289 
.4165 
.79U 

1.18 
.05 
.0489 

1.0622 


1.48 


EXCHANGE.  253 

WniTTJEN  EXERCISES. 

631. — Ex.  1.  How  much  must  be  paid  for  a  bill  on  Paris 
for  2565  francs,  exchange  being  at  5.13?  Ayis.  $500. 

2.  What  will  a  bill  cost  in  currency  on  Paris  for  10300 
francs,  exchange  being  at  5.15,  and  gold  at  120? 

3.  What  is  the  face  of  a  bill  on  Paris  that  can  be  purchased 
for  $2400  in  currency,  when  exchange  is  at  5.15,  and  gold  at 
120? 

4.  How  much  French  exchange  at  5.18  can  be  bought  for 
?  Ans.  3108  francs. 


TEST  QUESTIONS, 

632.— 1.  What  is  Stock?  What  is  the  par  vakie  of  stock?  The 
market  value  ?  When  does  gold  become  an  object  of  investment  ?  What 
is  a  broker  ? 

2.  What  is  a  Corporation?  What  is  a  corporate  stock?  A  share? 
An  installment  ?  An  assessment  ?  What  are  the  gross  earnings  of  a 
company?     The  net  earnings?     What  is  a  dividend  ? 

3.  Of  what  do  United  States  GovERNiMENT  Securities  consist  ?  What 
are  bonds?  Coupons?  Treasury  notes ?  According  to  what  are  the  bonds 
issued  by  cities,  counties,  states  and  corporations  named  ? 

4.  What  is  Exchange  ?  What  is  a  draft  ?  A  sight  draft  ?  A  time 
draft?     Who  is  the  drawer  of  a  draft?     The  drawee?     The  payee? 

5.  How  is  a  draft  accepted  ?  How  may  a  draft  be  assigned  ?  What 
kind  of  a  draft  is  entitled  to  grace?     What  is  the  course  of  exchange? 

6.  What  are  Inland  Bills?  What  is  the  base  of  a  bill?  The  rate? 
What  are  the  rules  for  inland  exchange  ? 

7.  What  are  Foreign  Bills?  W^hat  is  the  par  of  exchange  between 
two  countries  ?     How  are  foreign  bills  drawn  and  negotiated  ? 

8.  In  what  are  Bills  on  England  drawn  ?  By  what  is  the  value  of 
a  pound  sterling  represented  ?  To  what,  in  the  present  gold  coinage  of 
the  United  States,  is  a  sovereign  equal  ? 

9.  In  the  Computation  of  exchange  on  England,  what  value  of  a 
pound  is  considered  the  base?  What  is  the  value  of  exchange  when 
quoted  at  109?  When  quoted  at  109|?  What  are  the  rules  for  exchange 
on  England? 

10.  In  what  are  Bills  on  France  drawn?  How  may  francs  and 
centimes  be  written  ?  What  is  the  value  of  a  franc  in  silver  ?  How  is 
exchange  on  France  quoted? 

22 


254  GENERAL   TAXES. 


SECTION    LXII. 
GENERAL    TAXES. 

633.  Real  Estate  is  such  property  as  houses,  lands,  mills 
and  mines. 

634.  Personal  Property  is  movable  property,  as  money, 
stocks,  cattle  and  ships. 

635.  A  Tax  is  a  sum  of  money  imposed  on  persons  or  prop- 
erty for  public  purposes. 

636.  A  Property  Tax  is  a  tax  on  property,  and  is  reckoned 
at  a  certain  rate  per  cent,  on  the  estimated  value  of  the 
property. 

637.  An  Income  Tax  is  a  tax  on  an  income,  and  is  reckoned 
in  the  same  manner  as  a  property  tax. 

638.  A  Poll  Tax  is  a  tax  on  the  person  of  all  male  citizens 
not  exempt  by  law. 

In  some  States  no  poll  or  capitation  tax  is  imposed,  and  in  others,  as  in 
Vermont,  each  taxable  poll  is  reckoned  as  a  certain  amount  of  property. 
In  Massachusetts,  one-sixth  part  of  the  tax  to  be  raised  is  assessed  on  the 
polls,  provided  the  poll  tax  of  one  individual,  except  highway  taxes 
separately  assessed,  for  the  year  shall  not  exceed  $2. 

639.  Assessors  are  officers  appointed  to  estimate  the  value 
of  the  taxable  property,  to  make  a  list  of  taxable  polls,  if 
required,  and  to  apportion  the  tax  to  be  raised  among  the  tax- 
payers. 

640.  General  Taxes  are  such  as  are  imposed  for  city,  town, 
district,  county  or  state  purposes. 

WBITTJSy   EXERCISES. 

641. — Ex.  1.  A  tax  of  $21900  is  to  be  imposed  on  a  cer- 
tain town.  The  taxable  property  and  incomes  amount  to 
$1500000,  There  are  450  taxable  polls,  each  to  be  as- 
sessed $2.  Find  the  tax  of  S.  A.  Potter,  W.  B.  Rice,  and 
D.  J.  Snyder,  according  to  the  following — 


GENERAL   TAXES. 


255 


INVENTORY. 


Names. 

Real  Estate. 

Pek'l  Estate. 

Income. 

Total. 

S.  A.  Potter, 
W.  B.  Riee, 
D.  J.  Snyder, 

2 
1 
1 

$15000 
9300 

$31000 
12550 

$3160 
850 

$49160 

13400 

9300 

Solution.  . 
$2  X  4^0  —  ^900,  sum  to  he  assessed  on  the  polls. 
$21900  —  $900  —  $21000,  sum  to  be  assessed  on  valuation. 
$21000 -^ $1500000 ^ .014, Tate  of  taxation  on  valuation. 
$49160X  .014=  $688.24,  ^-  ^'  Potter's  tax  on  valuation. 
$688.24  +  ^4   =$692.24,  S.  A.  Potter's  entire  tax. 
$1S400X  .014  =  $187.60,  W.  B.  Rice's  tax  on  valuation. 
$187.60  +  $2    =  $189.60,  W.  B.  Rice's  entire  tax. 
$9800  X  .014    =$130.20,  D.  J.  Snyder's  tax  on  valuation. 
$130.20  +  $2    =$132.20,  D.  J.  Snyder's  entire  tax. 

2.  A  tax  of  $31000  is  imposed  on  a  certain  town.  The 
taxable  property  amounts  to  $3720000.  There  are  658  tax- 
able polls,  each  of  which  is  to  be  assessed  $1.50.  What  will 
be  the  tax  on  each  $1  of  valuation  ?  What  is  S.  M.  Allen's 
tax,  whose  valuation  is  $12500,  and  who  pays  for  one  poll  ? 

642.  Rule  for  General  Taxes.— Subtract  the  amoujit  of  the 
poll  taxes,  if  any,  from  the  whole  tax,  and  the  re- 
mainder ivill  be  the  tax  on  valuation. 

Apportion  the  tax  on  valuation  among  the  property 
owners  according  to  the  valuation  of  their  property, 
and  add  their  poll  tax,  if  any. 

The  computation  of  taxes  may  be  facilitated  by  constructing  a  table 
showing  the  tax  on  $1,  $2,  $3,  etc.,  from  which  to  compute  the  individual 
taxes. 


256 


GENERAL   TAXES. 


TABLE, 

For  a  Rate  of  IJf  Mills  on  a  Dollar. 


Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

Prop.         Tax. 

$1 

$.014 

$7 

$.098 

$40 

$.56 

$100 

$1.40 

$800  $11.20 

2 

.028 

8 

.112 

50 

.70 

200 

2.80 

900     12.60 

3 

.042 

9 

.126 

60 

.84 

300 

4.20 

1000     14. 

4 

.056 

10 

.14 

70 

.98 

500 

7.00 

2000     28. 

5 

.070 

20 

.28 

80 

1.12 

600 

8.40 

3000     42. 

6 

.084 

30 

.42 

90 

1.26 

700 

9.80 

4000     56. 

PHOBIjEMS. 

1.  Find  J.  Holt's  tax,  whose  valuation  is  $3060,  and  who 
pays  for  2  polls,  at  $1.50. 

Solution. 

Tax  hy  Table  on  $3000  =  ^42.00 

60=         .84 

"       $3060=  $42.84. 

Tax  on  2  polls  @  $1.S0  =       3.00 

Entire  tax  =  $45. 84- 

2.  Find  Walter  Hartman's  tax,  whose  valuation  is  S5350, 
and  who  pays  for  1  poll,  at  SI. 25. 

3.  A's  personal  estate  is  valued  $5600,  and  his  real  estate 
$5000.  What  is  his  tax,  if  the  rate  of  taxation  on  the  real 
estate  is  .015,  and  the  personal  estate  is  taxed  twice  as  high  as 
the  real  estate  ?  Ans.  $243.00. 

4.  A  tax  of  $6000  is  to  be  assessed  on  a  town  having  800 
polls  and  a  valuation  of  $500000.  If  one  sixth  of  the  tax  be 
laid  on  the  polls,  what  will  be  the  tax  on  $1,  and  how  nuioh 
will  be  each  poll  tax? 

5.  C.  Washburn  is  assessed  on  $5760  of  property  and  on  a 
taxable  income  of  $1240.  What  is  his  tax,  if  the  tax  on  $1 
of  valuation  is  $.016,  and  his  poll  tax  is  $2?    Ans.  $114.00. 


NATIONAL   TAXES.  257 

SECTION  LXIII. 
Ji'ATIOJ^AL  TAXES. 

643.  Customs,  or  Duties,  are  taxes  upon  imported  goods 
and  upon  the  tonnage  of  vessels. 

644.  An  Ad  Valorem  Duty  is  a  certain  per  cent,  on  the  net 
invoice  vahie  of  an  imported  article. 

645.  A  Specific  Duty  is  a  uniform  tax  on  certain  imported 
articles. 

646.  Tare  is  an  allowance  of  the  weight  of  a  cask,  bag  or 
case  containing  a  commodity,  and  which  has  been  weighed 
with  it. 

647.  Breakage  is  an  allowance  for  such  breakage  of  bottles 
which  contained  liquors  as  is  actually  ascertained  and  certified 
by  a  custom-house  appraiser. 

All  allowances,  in  general,  for  waste,  as  by  leakage,  are  such  as  may  be 
actually  ascertained.  But  no  allowance  is  made  for  the  loss  by  decay  of 
such  fruits  as  oranges,  lemons  and  bananas,  except  when  in  excess  of  25 
per  cent,  of  the  whole  quantity. 

648.  Internal  Revenue  is  the  revenue  of  Government  derived 
from  tax  on  incomes,  business  licenses,  stamps,  imposts  on 
manufactured  products,  etc. 

649.  National  Taxes  are  the  taxes  imposed  by  the  National 
Government. 

FTtO^IiETSlS. 

1.  What  is  the  duty,  at  2  cents  per  pound,  on  5400  pounds 
of  ginger-root,  tare  allowed  being  5%  ?  ^ws.'$102.60. 

2.  Lombard,  Brainard  &  Co.  imported  from  Havana  80 
hogsheads  of  molasses,  each  containing  68  gallons,  duty  5  cents 
per  gallon  ;  60  boxes  of  sugar,  each  weighing  450  pounds,  duty 
2f  cents  per  pound,  and  500  boxes  of  oranges,  invoiced  at  $1.50 
per  box,  duty  20%  ad  valorem.  Required  the  amount  of  duty 
on  the  whole.  Am.  $1144.50. 

3.  The  income  of  David  Welch   for   the  year   1870  was 

22* 


258  REVIEW  PROBLEMS. 

$6360.  Deductions  were  made  on  $2000  exempted  by  law, 
$500  for  house  rent,  $125  for  insurance,  $425  for  repairs,  $960 
for  labor  and  $200  for  taxes.  Required  the  national  tax  on 
the  balance  at  2|-%. 

4.  What  is  the  duty  on  500  dozen  bottles  of  champagne 
wine,  each  bottle  containing  1  quart  1  pint,  the  duty  at  $6  per 
dozen,  and  at  the  rate  of  $2  per  gallon  on  the  quantity  in 
excess  of  1  quart  per  bottle,  the  breakage  being  19  bottles  ? 

5.  Stockton  &  Bradley  have  imported  600  pounds  of 
prunes,  and  400  pounds  of  Zante  currants,  duty  on  each  2| 
cents  per  pound  ;  and  750  boxes  of  Sicily  oranges,  invoiced  at 
$4  per  box,  duty  20%  ad  valorem.  Required  the  duty  on  the 
whole,  provided  the  loss  of  the  oranges  by  decay  was  35%  of 
the  whole  quantity. 


SECTION   LXIV. 
REVIEW    PROBLEMS. 

MENTAJj    EXEJtCISES. 

650.— Ex.  1.  What  is  the  ratio  of  f  to  ^  ?     Of  |  to  2V  ? 

2.  Divide  42  into  two  parts  that  shall  be  in  the  ratio  of  f 
to  A. 

3.  Two  men  purchase  80  pounds  of  sugar.  One  pays  $2|-, 
and  the  other  $7^.     What  is  each  man's  share  of  it  ? 

4.  The  means  of  a  proportion  are  6  and  8.  One  of  the 
extremes  is  7.     What  is  the  other  extreme? 

5.  Three  men  hired  a  pasture  for  $108.  A  put  in  3  oxen, 
B,  4  oxen,  and  C,  5  oxen.     How  much  should  each  man  pay  ? 

6.  A  furnished  2  loaves  for  dinner,  and  B  furnished  3,  while 
C  contributed  30  cents  to  be  divided  between  A  and  B.  How 
much  should  each  receive  ? 

7.  A  man  wishing  to  contribute  money  to  an  equal  number 
of  poor  men  and  women,  gave  to  each  man  7  dimes,  and  to 
each  woman  5  dimes.  If  he  gave  them  in  all  $6,  how  many 
men  and  women  were  there  respectively? 


mi: VIEW    PROBLEMS.  259 

8.  Divide  $108  among  A,  B  and  C,  so  tliat  B  may  get  3 
times  as  much  as  A,  and  C  4  times  as  much  as  A. 

9.  A  and  B  hired  a  field  for  $96.  A  put  in  2  horses  for  3 
months,  and  B  put  in  2  horses  for  5  months.  How  much 
should  each  man  pay  ? 

10.  Two  men,  A  and  B,  traded  together.  The  contributioi- 
of  A  to  that  of  B  is  as  ^  to  \,  and  A's  money  Avas  in  the  busi- 
ness 3  months,  and  B's  4  months.  They  gained  $400.  AVhat 
Avere  their  respective  shares  of  it  ? 

11.  Bryant  owes  Forster  $50  payable  in  6  months,  and  $100 
payable  in  3  months.  What  is  the  average  time  for  paying 
the  whole  by  one  payment  ? 

12.  A  friend  has  loaned  me. $500  for  6  months.  How  long 
should  I  loan  him  $300  to  requite  the  favor  ? 

13.  The  rate  of  taxation  in  a  certain  toAvn  is  9  mills  on  a 
dollar  of  valuation.  What  is  the  rate  for  $100  of  valuation, 
and  Avhat  is  Johnson's  entire  tax,  whose  valuation  of  property 
is  $2000,  and  his  poll  tax  $2? 

14.  I  bought  50  shares  of  stock  at  97,  and  sold  it  at  101. 
What  did  I  gain,  alloAving  ^%  brokerage  on  each  transaction? 

15.  If  you  gain  $200  by  selling  50  shares  of  stock  for  101, 
at  what  rate  did  you  buy  the  shares  ? 

16.  Considering  the  value  of  a  pound  sterling  to  be  $4.86, 
and  the  value  of  5  francs  to  be  $.98,  how  much  less  do  25 
francs  differ  in  value  from  $5,  than  does  £1  ? 

WRITTEN   EXERCISES. 

651.— Ex.  1.  Which  is  the  greater  ratio,  7  :  22  or  113  :  355? 

Ans.  The  latter. 

2.  What  is  the  ratio  of  39  to  52,  expressed  in  its  lowest 
terras  ? 

3.  The  second,  third  and  fourth  terms  of  a  proportion  are 
17,  11  and  93|^.     What  is  the  first  terra  ? 

4.  A  man  purchased  a  horse,  a  cow  and  a  sheep  for  $205. 
The  cow  was  valued  at  as  much  as  10  sheep,  and  the  horse  at 
as  much  as  3  cows.     What  was  the  price  of  each  ? 

Am.  Slieep,  $5  ;  cow,  $50  ;  horse,  $150. 


260  iii:riEW  problems. 

5.  If  i|  of  a  dollar  will  pay  for  f  of  a  bushel  of  apples,  for 
what  part  of  a  bushel  will  /^  of  a  dollar  pay  ?         Ans.  ■^. 

6.  A  surveyor  having  measured  a  line  by  a  chain,  finds  it  to 
be  1584  yards,  but  on  examining  his  chain,  sees  it  deficient 
two  entire  links,  besides  several  not  straight — the  whole  error 
amounting  to  18  inches.  What  is  the  true  length  of  the  line, 
a  chain  measuring  66  feet  ?  Ans.  1548  yards. 

7.  If  18  men  can  construct  150  rods  of  road  in  25  days, 
how  many  men  will  be  required  to  construct  120  rods  in  15 
days? 

8.  If  15  bushels  of  corn  be  worth  as  much  as  2  barrels  of 
flour,  and  4  barrels  of  flour  as  much  as  5  tons  of  coal,  how 
many  bushels  of  corn  should  be  given  for  27  tons  of  coal  ? 

15  bu.  =  ^  hhl.  Solution.— By  arranging  the  terms  of  equal 

^  ])})l^  =:  ^  Jf.  value  so  that  denominations  of  the  same  kind 

2'y  rp  =  ^  jjj  shall   stand   on   opposite  sides  of   the  sign  of 

g       2  '  equality,  it  is  evident,  if  the  value  of  27  tons 

iSXJ-X  27  _  -lomi,       ^^^^^  known,  that  the  product  of  the  values  on 

-2-X-s-      ~    o^OU.    Qj-ig  gj^jg   Qf  ^j^g  gjgj^  would  be  equal   to   tlie 

product  of  those  on  the  other. 
Hence,  to  find  the  required  term,  divide  the  product  of  the  terms  on 
the  other  side  by  the  product  of  the  given  terms  on  its  own  side. 
This  method  is  sometimes  called  Conjoined  Proportion. 

9.  How  many  pounds  of  tea  must  be  given  for  28  pounds 
of  rice,  if  4  pounds  of  sugar  are  worth  1  pound  of  coffee,  15 
pounds  of  sugar  are  worth  14  pounds  of  rice,  and  30  pounds 
of  coffee  are  worth  7  pounds  of  tea  ?  Am.  If. 

10.  If  15  women,  working  12  hours  daily,  can  gather  60 
bushels  of  cranberries  in  16  days,  in  how  many  days  can  20 
boys,  working  10  hours  daily,  gather  98  bushels,  7  women 
being  able  to  do  as  much  as  8  boys  in  the  same  time  ? 

Am.  26f|. 

11.  If  $1  is  worth  4  s.  H  d.,  and  is  also  worth  5  francs 
17  centimes,  what  is  the  value  of  a  franc  in  sterling  money? 

12.  William  Hunt  owes  me  $3000,  payable  in  6  months. 
Should  he,  to  accommodate  me,  pay  one  third  of  the  amount 
down,  how  long  in  equity  should  he  be  permitted  to  retain  the 
remainder?  J ns.  9  months. 


EEVIEW   PROBLEMS.  261 

13.  A  and  B  are  in  partnersliip ;  A's  stock  of  $560  con- 
tinued in  trade  15  months,  and  drew  a  profit  of  $36 ;  B's  stock 
of  $320  was  in  trade  21  months.  What  should  be  the  amount 
•of  B's  gain  ?  Ans.  $28.80. 

14.  Bought  of  Messrs.  Bancroft,  Wright  &  Co.,  Sept.  1, 1870, 
a  bill  of  $756  on  4  months ;  Sept.  20,  a  bill  of  $144  on  2 
months ;  Oct.  3,  a  bill  of  $567  on  5  months ;  and  Oct.  16,  a 
bill  of  $128  on  6  months.  What  would  be  the  discount,  at 
6%,  on  a  note  for  the  amount,  dated  September  1,  and  matur- 
ing at  the  average  time  of  payment  ? 

15.  An  American  in  London  bought  £16000  consols,  an  En- 
glish government  security,  at  93f ,  and  sold  out  at  94|^.  What 
was  his  profit  in  United  States  money,  allowing  for  brokerage 
\(fo  on  each  transaction,  and  $4.86  as  the  value  of  a  pound 
sterling?  Ans.  $194.40. 

16.  What  will  be  the  cost  of  a  draft  of  $575  at  60  days, 
exchange  at  lOlf,  and  interest  at  7%  ? 

17.  W^hich  is  the  more  profitable  stock  for  investment — the 
U.  S.  4's,  at  85,  or  the  French  Rentes  3's,  at  65? 

Ans.  U.  S.  4's. 

18.  How  many  francs  are  worth  £1,  if  49^  d.  buy  $1,  and 
$200  bring  1034  francs  ? 

19.  At  what  must  gold  sell,  that  an  investment  in  U.  S.  5's, 
at  95,  nlay  yield  an  interest  of  6%  in  currency? 

Ans.  114. 

20.  For  what  price  must  a  10%  stock  sell  to  pay  an  interest 
of  8%  on  the  investment? 

21.  Sixty  gallons  of  alcohol  are  mixed  with  14  gallons  of 
water.  What  weight  of  alcohol  is  there  in  every  pound  of  the 
mixture,  the  weights  of  equal  measures  of  alcohol  and  water 
being  in  the  ratio  6:5? 

22.  The-  assessment  rolls  of  a  town  show  the  value  of  the 
taxable  property  to  be  $1000500,  and  the  number  of  polls  to  be 
600.  A  tax  of  $15207  is  to  be  raised.  What  Avill  be  the  rate 
on  property,  allowing  that  each  poll  shall  pay  $2  ?  What  is 
A's  entire  tax,  who  has  1  poll,  and  whose  property  is  valued 
at  $6500?  Ans.  Rate  on  property,  .014 ;  A's  tax,  $93. 


262  INVOLUTION. 

SECTION   LXV. 

I^rOLUTIOJ^. 

652. — Ex.  1.  What  is  the  result  of  taking  3  twice  as  a  factor  ? 

2.  AVhat  is  the  result  of  taking  5  twice  as  a  factor  ? 

3.  What  is  the  result  of  taking  2  three  times  as  a  factor  ? 

4.  What  is  the  result  of  taking  2  four  times  as  a  factor  ?    . 

5.  What  number  will  be  produced  by  taking  5  three  times 
as  a  factor  ? 

DEFINITIONS. 

653.  A  Power  of  a  number  is  the  result  obtained  by  using 
it  a  certain  number  of  times  as  a  factor. 

Thus,  16  is  a  power  of  4,  since  it  is  obtained  by  using  4  twice  as  a 
factor. 

654.  The  Powers  of  a  number  are  named  from  the  number 
of  times  it  is  used  as  a  factor. 

Thus,  4  is  the  second  power  of  2,  8  the  third  power,  and  1 6  the  fourth 
power,  since  2  is  used  twice  in  obtaining  the  first  number,  three  times  in 
obtaining  the  second,  and /our  times  in  obtaining  the  third. 

655.  The  second  power  is  frequently  called  the  Square,  from 
the  method  of  finding  the  area  of  a  square ;  and, 

The  third  power  is  frequently  called  the  Cube,  from  the 
method  of  finding  the  contents  of  a  cube. 

656.  The  Exponent  of  a  power  is  a  figure  or  figures  placed 
at  the  right  and  above  the  number,  to  show  the  number  of 
times  the  given  number  is  to  be  used  as  a  factor. 

Thus,  in  the  expression  3^,  2  is  the  exponent,  and  indicates  that  the 
square  of  3  is  to  be  found. 

The  powers  of  numbers  are  found  by  using  each  number  as 
many  times  as  a  factor  as  is  indicated  by  the  exponent. 

Any  number  that  is  the  product  of  equal  factors  is  called  a 
Perfect  Power. 

Any  number  that  is  not  the  product  of  equal  factors  is  called 
an  Imperfect  Power. 

657.  Involution  is  the  process  of  finding  a  power  of  a  num- 
ber. 


INVOLUTION.  263 

WRITTEN  EXEJiCISES. 

658.— Ex.  1.  Find  the  third  power  of  25. 

25  =  1st  power. 
25 
125 

50  Solution.— The  third  power  of  25,  or  25'', 

fjppr o  J  ^^  equal  to  the  product  of  25  taken  3  times  as 

^r  ~        P^'^^'r-  a  factor.     Hence,  15625,  the  equal  of  25  X  25 

"^^  X  25,  is  the  required  power. 


3125 
1250 

15625  =  3c?  poiver. 

2.  What  is  the  square  of  31  ? 

3.  What  is  the  cube  of  15?  '    Ans.  3375. 

659.  Rule  for  Involution.—  Use  the  ^iven  nwmber  as  many 
times  as  a  factor  as  is  indicated  by  the  exponent  of  the 
required  poiver. 

PnOBLEMS. 

1.  What  is  the  third  power  of  57  ?  Ans.  185193. 

2.  Raise  302  to  the  second  power. 

3.  What  is  the  square  of  962  ?  Ans.  925444. 

4.  Raise  407  to  the  fifth  power.        A7i3.  11167913618807. 

5.  Raise  f  to  the  fourth  power. 

Solution. 

"s^s^s^s^E'^^^  '^^^^'^^'^  power. 

6.  Raise  ^  to  the  third  power.  .  Ans.  ^l\l^. 

7.  What  is  the  value  of  (|)'?  Ans.  i^-i^. 

8.  What  is  the  fourth  power  of  1.05  to  four  orders  of  deci- 
mals ?  Ans.  1.2155. 

Here  all  the  figures  of  the  process  are  retained,  but  the  final  result  is 
expressed  approximately  only  to  the  nearest  decimal  of  the  fourth  order. 

9.  What  is  the  value  of  1.06^  to  three  orders  of  decimals  ? 

Ans.  1.338. 


264  EVOLUTION. 

SECTION    LXVI. 

EVOLUTIOJf. 

660.— Ex.  1.  What  are  equal  factors  of  25?  Of  49?  Of 
343? 

2.  Sixteen  is  the  result  of  taking  2  how  many  times  as  a 
factor  ? 

3.  One  hundred  twenty-five  is  the  result  of  taking  5  how 
many  times  as  a  factor? 

DEFINITIONS. 

661.  The  Root  of  a  number  is  one  of  the  equal  factors  of 
that  number. 

Thus,  8  is  a  root  of  64,  since  it  is  one  of  the  two  equal  factors  of  that 
number. 

662.  If  a  number  is  used  twice  as  a  factor  to  produce  a  cer- 
tain number,  it  is  called  the  second  or  square  root  of  that  num- 
ber ;  if  three  times,  the  third  or  cube  root ;  and  so  on. 

Thus,  4  is  the  second  or  square  root  of  16,  since  it  is  used  twice  as  a 
factor  in  producing  16 ;  the  third  or  cube  root  of  64,  since  it  is  used 
three  times  as  a  factor  in  producing  this  number. 

663.  The  roots  of  numbers  are  indicated  by  the  character, 
-[/,  called  the  Radical  Sign. 

If  no  figure  is  written  in  the  opening  of  the  sign,  the  square 
root  is  indicated ;  if  the  figure  3  is  placed  there,  as  -^y  the 
cube  root ;  if  4,  the  fourth  root ;  and  so  on. 

664.  An  Index  of  a  root  is  the  figure  or  figures  which  indi- 
cate the  root. 

No  imperfect  power  can  have  an  exact  root ;  and  a  root  that 
can  only  be  approximately  obtained  is  called  a  Surd  Boot,  or 
Surd. 

665.  Evolution  is  the  process  of  finding  tlie  roots  of  num- 
bers. 

The  roots  of  small  numbers  which  are  not  imperfect  powers 
may  be  readily  found  by  factoring. 


EVOLUTION.  266 

WltlTTEJf  EXERCISES. 

666.— Ex.  1.  Find  the  cube  root  of  3375. 
8)3375 
'3)1125 

3 J 37 5  Solution.— The  prime  factors  of  3375  are  3,  3,  3,  5,  5 

5)125       and  5. 

Since  the  cube  root  of  a  number  is  one  of  its  three 
equal  factors,  the  cube  root  of  3375  must  be  5  X  3,  or  15. 


5)25 
.     5 

5X3=15 

2.  Find  the  square  root  of  4096.  Ans.  64. 

3.  Find  the  cube  root  of  74088.  Ans.  42. 

4.  Find  the  fourth  root  of  1296.  Ans.  6. 

5.  Find  the  fifth  root  of  537824.  Ans.  14. 

GENEEAL  METHOD  FOR  SQUARE  ROOT. 

667.  A  General  Method  of  finding  the  square  root  of  num- 
bers may  be  deduced  from  raising  numbers  of  difierent  orders 
of  units  to  the  second  power,  and  observing  the  various  re- 
lations existing  between  the  roots  and  their  squares. 

668.  The  square  of  1  u,nit  of  units,  or  1,  is  1 ;  of  1  unit  of 
tens,  or  10,  is  100 ;  of  1  unit  of  hundreds,  or  100,  is  10,000 ; 
of  1  unit  of  thousands,  or  1000,  is  1000000 ;  and  so  on,  from 
which  it  may  be  seen  that 

The  square  of  every  number  between  1  and  10  must  be  a 
number  between  1  and  100 ;  of  every  number  between  10  and 
100  must  be  a  number  between  100  and  10000 ;  of  every 
number  between  100  and  1000  must  be  a  number  between 
10000  and  1000000  ;  and  so  on.     Hence, 

669.  The  square  of  a  number  of  units  must  be  found  either 
in  the  order  of  units,  or  in  tlie  orders  of  tens  and  units ;  the 
square  of  the  tens,  in  the  order  of  hundreds,  or  in  the  orders 
of  thousands  and  hundreds  ;  of  the  hundreds,  in  the  order  of 
ten-thousands,  or  in  the  orders  of  hundred-thousands  and  ten- 
thousands  ;  and  so  on.     Or, 

2S 


266  EVOLUTION. 

Numbers  with  1  figure  have  in  their  squares  1  or  2  figures, 
^      "  2  figures  "  "        ^  or  ^       " 

3  "  "  "        5  or  6       " 

4  "  "  "        7  or  8      " 
and  so  on. 

G70.  Take  any  number  composed  of  tens  and  units,  as  37, 
and  squaring  it,  as  in  actual  multiplication,  notice  of  what 
parts  the  square  is  composed  and  the  order  in  which  these 
parts  are  found.     Thus, 

37  =  3  tens  +  7  units 
3  tens  +  7  units 


3  tens  X  7  units  +  7  unit^ 
3  tens^  +  3  tens  X  7  units 

3  tens'  +  2X3  tens  X  7  units  +  7  units\ 
Or,  37^  =  30^  +  2X30X7 +7'-, 

hence,  37'  =  900  +  420  +  49  =  1369. 

Writing  t  for  tens  and  u  for  units,  and  omitting  the  figures 
indicating  the  number  of  the  tens  and  units,  we  have 

{t  +  uy  =  f  +  2XtXu  +  u\ 

Or,  more  concisely, 

{t  +  %iy=e  +  2tu  +  u\ 

671.  Principles. — 1.  The  number  of  figures  in  the  square 
root  of  a  number  is  one  half  of  the  number  of  figures  in  the 
square,  or  one  half  of  one  more  than  the  number  of  figures  in  the 
square. 

2.  If  the  square  of  an  integer  be  expressed  by  tivo  or  more 
figures,  the  first  two  orders  must  contain  the  square  of  the  unif^ ; 
the  next  two  higher  orders,  the  square  of  the  tens ;  the  next  two 
higher  orders,  the  square  of  the  hundreds;  and  so  on.     Hence, 

S.  If  a  square  be  separated  into  periods  of  two  figures  each, 
commencing  with  the  iinlts,  the  number  of  figures  in  the  root  is 


EVOLUTION.  267 

indicated,  and  also  the  orders  of  the  square  in  which  are  to  be 
found  the  squares  of  the  nmnhers  of  the  orders  of  the  root. 

4.  Every  square  consisting  of  more  than  two  orders  of  figures 
■is  equal  to  the  square  of  the  tens  in  its  root,  plus  twice  the  product 
of  the  tens  by  the  units,  plu^  the  square  of  the  units. 

Note.— The  square  of  a  fraction  canuot  be  integral,  or,  which  is  the  same  tiling,  an 
integer  cannot  have  a  fractional  square  root. 

No  exact  square  can  end  in  2,  3,  7,  8,  or  an  odd  number  of  ciphers. 

Terminating  decimals  can  only  be  squares  if  the  number  of  decimal  orders  be  even. 

WRITTEN  EXERCISES. 

672.— Ex.  1.  What  is  the  square  root  of  2209? 

t   n  Solution. — Since 

2209  =  <2  +  ^,„  +  „2  (_^  7  the  given  number  is 

4'^=  16        =  <^  expressed  by  4  or- 

— ; ders   of   figures,  it 

2t  =  2X4tens  =  S0)609=         Stu  +  u^  =  {2t  +  u^.  ^^^    be    separated 

— '^ —  into     two    periods, 

—  tf7  j^j^ J  jj.g  square  root 

"^^  ^  ^  =  609= {2t  +  ii)n     ^iii  consist  of  2  fig- 

Q  0       ures,  and  will   ex- 

press tens  and  units. 

Representing  the  tens  by  t  and  the  units  by  u,  we  have  2209  =  f  +  2<w 
+  ti^ 

In  reversing  the  process  of  involution,  we  must  first  find  the  f.  As  this 
must  be  a  number  of  hundreds,  we  find  the  greatest  number  contained  in 
the  left-hand  period  that  is  a  square  of  tens.  This  number  is  16  hun- 
dreds, whose  root,  4  tens,  we  write  for  the  tens  of  the  root. 

Subtracting  the  16  hundreds  from  the  given  number,  and  its  equal  t'' 
from  the  expression  above,  609  remains,  which  is  egual  to  2tu  +  u^ ;  or, 
since  u  is  a  common  factor  of  these  terms,  equal  to  {2t  -\-  u)  u. 

Of  these  two  factors,  2J  -f  m  and  w,  2t  is  the  only  term  whose  value  we 
can  find ;  if  tlie  whole  of  either  could  be  found,  we  should  use  that  factor  in 
finding  the  other.  Since  2<  of  the  factor  2<  +  u  is  large  compared  Avith 
the  other  term,  we  u.se  it  as  a  trial  divisor  in  finding  the  value  of  u. 

Now,  2<  =  2  X  4  tens  =  8  tens;  and  dividing  the  60  tens  of  the  re- 
mainder by  the  8  tens,  we  have  7,  which  we  write  for  the  units  of  the  re- 
quired "root. 

Adding  to  80,  the  equal  of  2t,  7,  the  equal  of  u,  we  have  87, 
and  multiplying  this  by  7,  the  equal  of  u,  we  obtain  609.  Subtracting 
this  product  and  its  equal  (2f  -\-u)  u  from  the  quantities  above  them. 
We  find  no  remainder.     Hence,  the  square  root  of  2209  is  47. 


t 


268  EVOLUTION. 

2.  What  is  the  square  root  of  55225  ? 
Solution. 

t   X(,   u 

562\25  =  t^  +  2(u  +  xfl(235 

£t  =  SX2teus  =  40jl52  =  S(u  +  u^  =  {St  +  u)u 

«  =3 

2t  +  u  =43 

43X3=         129         =_ i^t.  +  u)u 

2i  =  2X23 tens  =  46O  J 23 25  =  Stu  +  ifi  =  {2t  +  u)u 

u     =      5 

St  +  u  =465 

465  X  5  =  2325  = i^t  +  ^'> 

0  0 

Since  the  given  number  is  expressed  by  5  orders  of  figures,  it  may  be 
separated  into  three  periods,  and  its  square  root  will  consist  ot  3  figures, 
and  will  express  hundreds,  tens  and  units. 

Eepresenting  the  hundreds,  or  tens  of  tens,  by  t,  and  the  units  of  the 
tens  by  u,  the  square  of  the  root  thus  represented  will  equal  (-  +  2tu  + 
u^.     This  root,  then,  will  be  found  in  552  units  of  hundreds. 

The  greatest  number  contained  in  the  left-hand  period  that  is  a  square 
of  tens  is  4  hundreds,  whose  root,  2  tens,  we  write  for  the  tens  of  tens  of  the 
root.  Subtracting  4  hundreds  and  its  equal  f  from  the  expression  above, 
152  hundreds  remain,  and  2ti(,  +  u\  which  is  equal  to  (2<  +  u)u. 

Since  2t  of  the  factor  2t  +  u  is  large  compared  with  the  other  term,  we 
use  it  as  a  trial  divisor  in  finding  the  value  of  u.  Now,  2i  =  2  X  2  tens 
=  4  tens,  and  dividing  the  15  tens  of  the  remainder  by  the  4  tens,  we 
have  3,  which  we  write  for  the  units  of  the  tens  of  the  root. 

Adding  to  2t  or  40,  3,  the  equal  of  u,  we  have  43,  and  multiplying  this 
by  3,  the  equal  of  w,  we  obtain  129.  Subtracting  129  from  the  dividend, 
and  its  equal  {2t  -{-u)u  from  the  expression  above  it,  and  bringing  down 
the  remaining  period  of  the  power,  we  have  as  a  remainder  2325. 

We  now  consider  55225  as  the  square  of  23  tens  and  some  number/)f 
units  of  units.  Eepresenting  the  23  tens  of  the  root,  which  have  been 
found,  by  t,  and  the  \inits  of  the  root  to  be  found,  by  n,  the  square  of  tlie 
root  is  represented  by  <*  +  2lu  +  u'^.  The  equal  of  t"^,  or  230^,  has  been  sub- 
tracted ;  hence,  the  remainder,  2325,  is  equal  to  2tu  +  u";  or  (2/  +  u)u. 

We  find  the  value  of  2t  to  be  46  tens,  or  460,  and  making  use  of  it  as 
a  trial  divisor  for  finding  the  value  of  u,  we  have  for  that  value  5,  which 
we  write  for  the  units  of  the  root. 

Adding  to  460,  the  equal  of  2/.  5.  the  equal  of  w,  and  multiplying  this 
by  5,  the  equal  of  u,  we  obtain  2325.     Subtracting  2325  from  the  divi- 


E  VOL  VTION.  269 

dend,  and  its  equal  {2t-{-u)u  from  the  expression  above  it,  we  find  no 
remainder.     Hence,  tlie  square  root  of  55225  is  235. 

If  the  root  had  consisted  of  four  or  more  figures,  the  explanation  of 
the  process  of  solution  would  have  been  similar.  Thus,  if  the  root  had 
consisted  of  four  figures,  expressing  thousands,  hundreds,  tens  and  units, 
we  should  represent  tlie  thousands  or  tens  of  hundreds  by  /,  and  the  units 
of  hundreds  by  u,  and  having  found  their  values,  should  regard  the  part 
of  the  root  found,  as  tens  of  tens,  and  represent  it  by  t,  and  the  units  of 
tens  by  u,  and  so  on. 

If  the  given  number  had  been  a  decimal  fraction,  its  periods  would 
have  been  pointed  off  to  the  right,  beginning  with  units;  for  the  square 
of  .1  is  .01,  the  square  of  .09  is  .0081,  etc. 

3.  Find  the  square  root  of  4096.  Ans.  64. 

4.  Fiud  the  square  root  of  64516.  Ans.  254. 

673.  Rule  for  Square  ^oo\.— Point  off  the  given  munher 
into  periods  of  two  orders  each,  beginning  with  the 
units,  and  proceeding  toward  the  left  a.nd  right. 

Find  the  greatest  square  in  the  highest  period,  con- 
sidered as  units,  and  place  its  root  at  the  right  for  the 
first  figure  of  the  required'  root.  Subtract  this  square 
from  the  highest  period,  and  to  the  remainder  bring 
down  the  next  period  for  a  dividend- 

Tahe  for  a  trial  divisor  twice  the  root  already  found, 
considered  as  tens ;  divide  the  dividend,  omitting  its 
right-hand  order,  by  the  trial  divisor,  and  ivrite  the 
quotient  for  the  second  figure  of  the  required  root. 

To  the  trial  divisor  add  the  part  of  the  root  found 
by  it,  multiply  the  result  by  that  part  of  the  root,  and 
subtract  the  product  froiyv  the  dividend. 

Continue  the  process,  if  there  are  other  periods,  as 
before. 

The  trial  divisor  being  less  than  the  true  divisor,  the  probable  root 
found  by  it  may  prove  too  large ;  if  so,  diminish  by  1  or  more,  and  re- 
new the  process. 

When  0  occurs  in  the  root,  instead  of  indicating  the  multiplication  by 
0  and  subtracting,  it  is  simpler  to  annex  a  cipher  to  the  trial  divisor, 
and  to  the  dividend  bring  down  another  period. 
28* 


270  EVOLUTION. 

If  there  be  a  remainder  after  all  the  periods  have  been  used,  periods 
of  decimals  may  be  formed  by  annexing  ciphers,  and  the  work  continued. 

PROBLEMS. 

What  is  the  square  root — 

1.  Of  9216  ?  Ans.  96. 

2.  Of  4096  ? 

3.  Of  .0676  ?  Am.  .26. 

4.  Of  717409  ?       Am.  847. 


5.  Of  46656?  Ans.  216. 

6.  Of  7569  ? 

7.  Of  62504836  ?  Ans.  7906. 

8.  Of  21.16  ?  Ans.  4.6. 


9.  Extract  the  square  I'oot  of  5  to  three  decimal  orders,  or 
to  within  less  than  yuVo-  -^^s.  2.236. 

10.  Extract  the  square  root  of  .5  to  four  decimal  orders,  or 
to  within  less  than  ^q^qq.  Ans.  .7071. 

11.  A  general  has  11664  men.  How  many  must  he  place 
in  rank  and  file  to  form  them  into  a  square  ?  Ans.  108. 

12.  A  square  lot  contains  18225  square  feet.  What  is  the 
length  of  its  equal  sides  ?  Ans.  135  feet. 

13.  What  is  the  value  of  i/.00062o  ?  Ans.  .025. 

14.  A  circular  garden  contains  6561  squar.e  feet.  What  is 
the  length  of  one  side  of  a  square  containing  the  same  number 
of  square  feet  ? 

15.  A  man  wishes  to  lay  out  a  farm  in  a  square  containing 
exactly  140  acres  100  square  rods.  What  must  be  the  length 
of  each  side?  Ans.  150  rods. 

674.  Since  the  square  of  a  fraction  is  found  by  squaring  the 
numerator  and  denominator,  the  square  root  of  a  fraction  is 
found  hy  taking  the  sqiuire  root  of  the  numerator  and  denom- 
inator. 

16.  What  is  the  square  root  of  ||^  ? 

^  224       16        fie       4    A^o 

feoLVT.ox.-^-^-;  ^~  =  l,Ans. 

17.  Find  the  square  root  of  ^^^.  Ans.  f^. 

18.  Find  the  square  root  of  aVA-  -^^^^^  t^- 

19.  Find  the  square  root  of  lljf.  Aiu.  y,  or  3f. 

20.  What  is  the  vahie  of  i  30]:?  Ans.  y,  or  5^. 


EVOl^UTION.  271 

675.  When  the  numerator  and  denominator  of  a  fraction 
are  not  both  squares,  to  find  the  approximate  square  root, 

First  reduce  the  fraction  to  a  decimal,  and  then  tahe 
the  root.     Or, 

Multiply  the  momerator  by  the  denominator,  and 
divide  the  root  of  the  product  hy  the  denominator. 

21.  What  is  the  square  root  of  f,  to  within  less  than  y^? 


Solution.-^  =  .4.285  +  I  V .4-285  =  .65 +. 


Or, 


V5_     \SX7  _     III.— 4:58  __   ^^ 
7~\'n     ~\49  ~     7     --^^ 


+  . 


22.  Find  the  square  root  of  ^,  to  within  less  than  y oVo- 

Ans.  .807. 

23.  What  is  the  square  root  of  ff  |-,  to  within  less  than  yoVo  ? 

24.  What  is  the  square  root  of  Gf,  to  within  less  than  yowo  ^ 

GENERAL  METHOD  FOR  CUBE  ROOT. 

676.  A  General  Method  of  finding  the  cube  root  of  num- 
bers may  be  deduced  from  raising  numbers  of  different  orders 
of  units  to  the  third  power,  and  noticing  the  various  relations 
existing  between  the  roots  and  their  cubes. 

677.  The  cube  of  1  unit  of  units,  or  1,  is  1 ;  of  1  unit  of  tens, 
or  10,  is  1000;  of  1  unit  of  hundreds,  or  100,  is  1000000;  of 
1  unit  of  thousands,  or  1000,  is  1000000000;  and  so  on.   Hence, 

The  cube  of  units  must  be  found  in  the  orders  of  units,  tens 
and  hundreds,  or  in  the  units'  period ;  of  tens,  in  the  orders 
thousands,  ten-thousands  and  hundred-thousands,  or  in  the 
thousands'  period  ;  of  hundreds,  in  the  millions'  period ;  of 
thousands,  in  the  billions'  period,  and  so  on.     Or, 

Numbers  with  1  figure  have  in  their  cubes  1,  2  ov  3  figures. 

2  figures         "  "  4,5  or  6       " 

3  figures         "  "  7,  8  or  9       " 

4  figures        "  "     10,  11  or  12    " 
and  so  on. 


272  EVOLUTION. 

678,  Take  any  number  composed  of  tens  and  units,  as  47, 
and  cubing  it  as  in  actual  multiplication,  notice  of  what  parts 
the  cube  is  composed,  and  the  orders  in  which  these  parts  are 
found.     Thu5, 

^7  =  4  ^^^^  "^  ^  wwifs 

^  tens  +  7  units 

Jj,  tens  X  7  units  +  7  ^<mfe^ 

4  ^e/is^  +  4  i^^  X  7  units 

4 tens^  +  2X4 i^''^ X  '^ '^^■'■^  +  '^ units^ 
4  tens  +  7  «?riVs 

^  fens*  X  7  wmfe  +  ^  X  ^  tew^  X  7  w?iife''  +  7  w?ir'is^ 

4  ten^  +  2x4 ^1^ X  7 itm'fe  +  ^  fern  X  7  t^m'^' 

^  tens^  +3x4  tens''  X  7  units  +  SX4tensX7  units'"  +  7  U7iits^ 
Writing  t  for  tens,  and  u  for  units,  and  omitting  the  figures  in- 
dicating the  number  of  the  tens  and  units,  we  have 

(t-{-uf  =  f-i-3XfXu+SXtX  u'  +  u\ 
Or,  more  concisely, 

{t  +  uf  =  f  +  seu  4-  Otu?  +  U\ 

679.  Principles. — 1.    The  number  oj  figures  in  the  cube  root 
^  of  a  number  is  one  third  of  the  number  of  figures  in  the  cube,  or  one 

third  of  one  or  tivo  more  tlian  the  number  of  the  figures  in  the  cube. 

2.  If  the  cube  of  an  integer  be  expressed  by  three  or  more 
figures,  the  first  three  orders  must  contain  the  cube  of  the  units ; 
the  next  three  higher  orders,  the  cube  of  the  tens,  the  next  three 
higher  orders,  the  cube  of  the  hundreds,  and  so  on.     Hence, 

3.  Jf  a  cube  be  separated  into  periods  of  three  figures  each, 
commencing  with  the  unih,  the  number  of  figures  in  the  root  is 
indicated,  and  also  the  orders  of  the  cube  in  which  are  to  be  found 
the  cubes  of  the  numbers  of  the  given  orders  of  the  root. 

4.  Every  cube  consisting  of  more  than  three  orders  of  figures 
is  equal  to  the  cube  of  the  tens,  plus  three  times  the  product  of  the 
square  of  the  tens  by  the  ^lnits,  plus  three  times  the  product  of  the 
tens  by  the  .'square  of  the  units,  plus  the  cube  of  the  units. 

Note — The  cube  of  a  fraction  cannot  be  intognil. 

A  cnbfi  niiiy  be  an  expression  endinR  in  any  fipnro.  but  if  the  expression  end  in  0,  it 
must  en<l  in  a  number  of  ciphers  divisible  by  3. 
Teruiinntiiig  decimals  can  only  be  cubes  if  the  number  of  decimal  orders  be  divisible  by  3. 


EVOLUTION.  273 

WRITTEN    EXERCISES. 

680.— Ex.  1.  What  is  the  cube  root  of  300763? 
Solution. 

t  u 

300763  =  <3  +  sfiu  +  stu^  ■vv?(G  7 
6'=  216        =<' 


SP  =  Sy.6iens>'     =10800  )  84-763     =         Sf^u-V8tu^-^v?  = 

Stu  =  SX6tensX7=   1260  {Sf  +  Stu  +  m^jm 

3f^  +  3tu  +  w2         =  12109 

12109  X  7  =  84763      =         JSf^  +Stu  +m')m 

0  0 

Since  the  given  number  is  expressed  by  6  orders  of  figures,  it  may  be 
separated  into  2  periods,  and  its  cube  root  must  consist  of  2  figures,  ex- 
pressing tens  and  units. 

Eepresenting  the  tens  by  t,  and  the  units  by  n,  300763  is  represented 
by  t^  +  sC-u  +  Stv?  +  V?. 

In  reversing  the  process  of  invohition,  we  must  first  find  l^,  and  as  this 
must  be  a  number  of  thousands,  we  find  the  greatest  number  contained 
in  the  left-hand  period  that  is  a  cube  of  tens.  Tliis  number  is  216  thou- 
sands, Avhose  root  is  6  tens,  which  we  write  for  the  tens  of  the  required 
root. 

Subtracting  216  thousands  from  the  cube,  and  its  equal,  t^,  from  the 
expression  above,  84763  remains,  equal  to  Zt'^u  +  3te^  +  u^,  or,  since  u  is 
a  common  factor  of  these  terms,  equal  to  (3<^  +  Ztu  +  m^)m. 

Of  the  two  factors  of  the  latter  expression,  one  terra,  3<^  of  the  factor 
within  the  parentheses  can  be  found ;  and  since  this,  compared  with  the 
other  terms,  is  large,  we  find  and  make  use  of  its  equal  as  a  trial  divisor 
for  finding  the  equal  of  the  factor  u.  Now,  3<^  =  3  times  6  tens  square, 
or  108  hundreds ;  and  dividing  the  847  hundreds  of  the  remainder  by 
the  108  hundreds,  we  have  7,  which  we  write  for  the  units  of  the  required 
root. 

Adding  to  the  108  hundreds,  126  tens,  the  equal  of  Ztu,  and  49,  the 
equal  of  w^,  we  have  12109 ;  and  multiplying  this  by  7,  the  equal  of  u, 
we  obtain  84763,  which  we  write  equal  to  (3<^  -f  3<w  +  v?)u.  Subtracting 
these  quantities  from  those  above  them,  we  find  no  remainder.  Hence, 
the  cube  root  of  3D0763  is  67. 

2.  What  is  the  cube  root  of  658503  ? 

3.  What  is  the  cube  root  of  970299  ?  Ans.  99. 


274  EVOLUTION. 

4.  Find  the  cube  root  of  24137569. 

Solution.  ,_^ 

24,1371669  =  t^  +  3Pu  +  stu^  +  u^(2  8  9 
2^=  8  =t^ 

8f^  =  3X2tens^     =    1200  )  16 137  =         8f^u  +  8tu^  +  v?  = 

Stu  =  SX2tensX8=     480  {8t^  +  8tu  +  u^)u 

u^=8^  =       64 

8t^+8tu  +  u^         =    1744 

1744  ^8  =  13952  =  {Sf'  +  Stu  +  u^)u 

3P  =  SX 28 tens-i    =  235200        )   2185569  =  {3t^  +  3tu  +  m2)m 

3tu  =  3X28tensX9=     7560 
u^  =  9^  =        81 


3f^-\-3tu  =  ui         =242841 

242841X9  =  2185569   =  {St^  +  3tu  +  u-^)u 

0  0 

Since  the  given  number  is  expressed  by  8  figures,  it  may  be  separated 
into  3  periods,  and  its  cube  root  must  consist  of  3  figures,  expressing 
hundreds,  tens  and  units. 

Eepresent  the  hundreds,  or  tens  of  tens  of  the  root,  by  t,  and  the  units 
of  tens  by  u.  The  cube  of  the  root  thus  represented  must  equal  t^  + 
St^w  +  3«M^  +  u*.  This  root,  then,  will  be  found  in  24137  units  of 
thousands. 

The  greatest  cube  of  any  number  of  tens  contained  in  24  thousands  is 
8  thousands,  whose  root,  2  tens,  we  write  for  the  tens  of  tens  of  tlie  re- 
quired root. 

Subtracting  8  thousands  from  the  24  thousands,  and  its  equal,  fi,  from 
the  expression  above,  16137  remains,  Avhich  is  equal  to  St-u  +  Stii^  +  u% 
or,  since  ia  is  a  common  factor  of  these  terms,  16137  must  contain  the 
product  of  (3<^  +  3tu  -\-  u^)  by  u. 

We  find  the  value  of  St'^  to  be  12  hundreds,  and  make  use  of  it  as  a 
trial  divisor  for  finding  the  value  of  u.  Dividing  the  161  hundreds  of 
the  remainder  by  the  12  hundreds,  we  have  13,  or  1  ten  and  3  units. 
But  as  we  have  already  found  and  subtracted  tlie  cube  of  tlie  greatest 
number  of  tens,  we  know  that  the  quotient  is  too  large  (as  may  often  be 
tlie  case,  since  tlie  trial  divisor  is  much  smaller  than  the  factor  of  which 
it  is  only  a  term),  and  tlverefore  diminish  it  to  9.  Taking  that  as  the 
I)robable  number  of  units  of  tens  of  the  required  root,  add  to  the  trial 
divisor  04  tens  and  81  imits,  the  equals  of  3tii,  and  h'',  and  we  have  1821. 
But  1821  multiplied  by  9  equals  16389,  wliich  is  greater  than  the  divi- 
dend.     The   quotient  is  therefore  still  too  large,  and  we  diminish  it 


EVOLUTION.  275 

by  1,  and  have  8,  wliicli  we  write  for  tlie  units  of  tens  of  the  required 
root. 

Adding  to  the  trial  divisor  48  tens  and  64,  the  equals  of  3te  and  u^,  the  . 
sum  is  1744,  and  this  multiplied  by  8  equals  13952.     Subtracting  13952 
from  the  dividend,  and  its  equal   (3f-  +  3<«  +  ?(.')«  from  the  expression 
above  it,  and  bringing  down  the  i-emaining  period  of  the  cube,  we  have  a 
remainder  of  2185569. 

Regarding  the  tens  and  units  of  tens  already  found  as  28  tens  of  the 
order  of  units,  we  represent  them  by  t,  and  the  imits  which  are  to  be 
found  by  u.  The  cube  of  the  root  now  represented  must  equal  t^  -\-  St''u 
-\-  Stu'^  +  u^.  The  cube  of  the  28  tens  here  represented  by  t?  has  been 
already  subtracted  from  the  given  cube;  hence  the  remainder,  2185569, 
must  equal  Sf'u  +  Slu^  +  u^,  or  {Se  +  3iu  +  u'^)u. 

We  find  the  value  of  3<^  to  be  2352  hundreds,  and  make  use  of  it  as  a 
trial  divisor  for  finding  the  value  of  n.  Dividing  the  21855  hundreds 
of  the  remainder  by  the  2352  hundreds,  we  have  9,  which  we  write  as 
the  units  of  the  required  root. 

Adding  to  the  trial  divisor  756  tens  and  81,  the  equals  of  3tu  and  u'\ 
■we  have  242841 ;  and  multiplying  this  by  9,  the  equal  of  u,  we  obtain 
2185569,  which  we  write  equal  to  (3<^  +  Stu  -\-  u'^)u.  Subtracting  these 
equals  from  the  equals  above,  we  have  no  remainder.  Hence,  the  cube 
root  of  24137569  is  289. 

If  the  given  number  had  been  a  decimal  fraction,  its  periods  would 
have  been  pointed  off,  beginning  with  units,  to  the  right;  for  the  cube 
of  .1  is  .001,  the  cube  of  .09  is  .000729,  etc. 

5.  Find  the  cube  root  of  91125000.  Ans.  450. 

6.  Find  the  cube  root  of  34012.224.  Ans.  32.4. 

681.  Rule  for  Cube  Root— Point  ojf  the  given  niunher 
into  periods  of  three  orders  each,  beginning  with  the 
units,  and  proceeding  toward  tlie  left  and  right. 

Find  the  gj^eatest  cuhe  in  the  highest  period,  con- 
sidered as  units,  and  place  its  root  at  the  right  for 
the  first  figure  of  the  root.  Suhtract  this  cuhe  from 
the  highest  period,  and  to  the  remainder  bring  down 
the  next  period  for  a  dividend. 

Tahe  for  a  trial  divisor  three  times  the  square  of 
the  part  of  the  root  already  found ;  divide  the  divi- 
dend, omitting  its  two  right-hand  orders,  and  ivrite 
the  quotient  for  the  second  figure  of  the  required  root. 


276  EVOLUTION. 

To  the  tibial  divisor  add  three  times  the  product  of 
the  preceding  part  of  the  root  (considered  as  tens)  hy 
the  last  root  figure,  and  also  the  square  of  the  last 
figure;  multiply  the  sutyi  by  the  last  root  figure,  and 
subtract  the  product  frojn  the  dividend. 

Continue  the  process,  if  there  are  other  periods,  as 
before. 

The  quotient  found  by  the  trial  divisor,  since  that  divisor  is  less  than 
the  true  divisor,  must  often  be  diminished  by  1  or  more. 

When  0  occurs  in  the  root,  annex  two  ciphers  to  the  trial  divisor,  and 
to  the  dividend  another  period. 

If  there  be  a  remainder  after  all  the  periods  have  been  used,  periods 
of  decimals  may  be  formed  by  annexing  ciphers,  and  the  ■work  con- 
tinued. 

What  is  the  cube  root — 


1.  Of  110592?  Arts.  48. 

2.  Of  91125? 

3.  Of  178453.547?  Am.  56.3. 


4.  Of  373248?  Am.  72. 

5.  Of  157464? 

6.  Of  1003003000? 


7.  Extract  the  cube  root  of  379,  to  withiu  less  than  -nyVo"- 

Am.  7.236. 

8.  Extract  the  root  of  .08,  to  within  less  than  ^(^^q,;). 

Am.  .4308. 

9.  What  is  the  length  of  one  side  of  a  cubical  box  con- 
taining 42875  cubic  inches  ?  Ans.  35  inches. 

10.  What  is  the  side  of  a  cube  which  will  contain  as  much 
as  a  bin  8  feet  3  inches  long,  3  feet  wide,  and  2  feet  7  inches 
deep  ?  Ans.  47.98  +  inches. 

11.  The  capacity  of  a  cubical  cistern  is  74088  cubic  inches. 
What  is  the  area  of  its  battom?      Am.  1764  square  inches. 

12.  Required  the  dimensions  of  a  cube  that  shall  have  the 
capacity  of  a  chest  2  feet  8  inches  long,  2  feet  3  inches  wide, 
and  1  foot  4  inches  thick.  An?.  24  inches. 

682.  Since  the  cube  of  a  fraction  is  found  by  cubing  the 
numerator  and  denominator,  the  eiibe  root  of  a  fraction  is  found 
by  taking  the  ctibe  root  of  the  numerator  and  denominator. 


EVOLUTION.  277 

13.  Wliat  is  the  cube  root  of  fl^? 

Solution. — ^  77  =  ~r,  =  7* 

\  ^4      f64      4 

14.  What  is  the  cube  root  of  ^V^  ?  ^Ins.  -^. 

15.  What  is  the  value  of  ifHil?  ^ns.  ff. 

16.  What  is  the  cube  root  of  49^?  Ans.  3|. 
683.  When  the  numerator  and  denominator  of  a  fraction 

are  not  both  cubes,  to  find  the  approximate  cube  root, 

First  reduce  the  fraction  to  a  decimal,  and  the?i  take 
the  root. 

17.  What  is  the  cube  root  of  f,  to  within  less  than  y-gVo"^ 

Solution. 
j ^.571428571+;  -^.57 1428571  =  . 829,  Aiw. 

18.  What  is  the  cube  root  of  -^^,  to  within  less  than  ytj-^tj-jt^ 

Ans.  .2714. 

19.  What  is  the  cube  root  of  1\,  to  within  less  than  y^? 

Ans.   1.93. 
20.  What  is  the  length  of  the  edge  of  a  cubical  block  of 
marble,  to  within  ytq-o,  if  the  block  contains  9|-  cubic  feet  ? 

Am.  2.092  feet. 


TEST    QUESTIONS. 

684. — 1.  What  is  a  Power?  From  what  are  the  powers  of  a  num- 
ber named  ?  What  is  the  second  power  frequently  called  ?  The  third 
power  ?     What  is  the  exponent  of  a  power  ? 

2.  What  is  Involution  ?  By  what  is  involution  performed  ?  Ex- 
plain how  a  number  is  raised  to  the  third  power.  What  is  the  rule  for 
involution  ? 

3.  What  is  Evolution?  What  is  a  root  of  a  number?  What  is  the 
second  or  square  root  of  a  number  ?  The  third  or  cube  root  ?  What  is 
the  sign  called  by  which  the  roots  of  numbers  are  indicated  ?  What  is 
the  index  of  the  root  ?  What  is  a  surd  ?  How  may  the  roots  of  small 
numbers,  which  are  not  imperfect  powers,  be  readily  found  ? 

4.  What  is  the  Kule  for  Square  root?     For  Cube  root? 

24 


278  MENSURATION. 

SECTION    LXVII. 

MEJfSUBATIOjY. 

685.  A  Point  is  that  which  has  only  position. 

686.  A  Plane  is  a  surface  (Art.  302)  in  which,  any  two 
points  being  taken,  the  straight  line  joining  them  will  be 
wholly  in  the  surface. 

687.  A  Curyed  Line  is  one  that  has  no  part  in  a  straight  line 
(Art.  308). 

688.  Parallel  Lines  are  such  as  are  wholly  in  the  same  plane 
and  have  the  same  direction. 


A  Curved  Lime.  Parallel  Lines. 

689.  A  Right  Angle  is  an  angle  formed  by 
one  straight  line  meeting  another,  making 
the  adjacent  angles  equal. 

The  straight  line  meeting  the  other  is  said 
to  be  a  perpendicular  to  it.  

Two  Right  Angles. 

690.  An  Acute  Angle  is  less  than  a  right  angle. 

691.  An  Obtuse  Angle  is  greater  than  a  right  angle. 


An  Acute  Angle.  An  Obiuse  Angle. 

692.  Mensuration  treats  of  the  measurement  of  lines,  sur- 
faces and  volumes  (Art.  303). 

POLYGONS. 

693.  A  Triangle  is  a  plane  bounded  by  three  straight  lines. 

694.  An  Equilateral  Triangle  has  the  three  sides  equal. 


MENSURA  TIOK. 


279 


695.  An  Isosceles  Triangle  has  two  equal  sides. 
690.  A  Scalene  Trian8:le  has  three  unequal  sides. 

697.  A  Hi  gilt -angled  Triangle  has  one  right  angle. 
The  side  opposite  the  right  angle  is  called  the  hypotenuse. 

The  side  meeting  another  side  and  forming  the  right  angle  is  called 
the  perpendicular. 

698.  An  Acute-angled  Triangle  has  three  acute  angles. 

699.  An  Obtuse-angled  Triangle  has  one  obtuse  angle. 


A  RlQHT-ANGLED  TRIANGLE.       AN  ObTDSE-ANGLED  TRIANGLE.        AN  AcniE-ANGLED  TRIANGLE. 

700.  A  Quadi'ilateral  is  a  plane  bounded  by  four  straight 
lines. 

701,  A  Parallelogram  is  a  quadrilateral  having  its  opposite 
sides  parallel. 

703.  A  Rhomboid  is  a  parallelogram  having  no  right  angles. 
703.  A  Kliombus  is  a  rhomboid  having  all  its  sides  equal. 


A  Parallelogram. 


A  Rhomboid. 


704.  A  Trapezoid  is  a  quadrilateral  having  only  two  of  its 
sides  parallel. 

705.  A  Trapezium  is  a  quadrilateral  having  no  two  of  its 
sides  parallel. 

706.  A  Diagonal  is  a  straight  line  which  joins  two  angles 
of  a  plane  figure  which  are  not  adjacent. 

707.  The  Base  of  a  figure  is  the  side  upon  which  it  is  sup- 
posed to  stand. 


A  Trapezoid. 


A  Trapezium. 


280  MENSURATION. 

708.  A  Polyg-on  is  a  plane  figure  bounded 
by  straight  lines. 

709.  A  Regular  Polygon  has  all  its  angles 
and  all  its  sides  equal. 


A  Regular  Polygon. 


Of  regular  polygons,  the — 
Pentagon  has  5  sides. 
Hexagon  has  6  sides. 
Heptagon  has  7  sides. 


Octagon  has  8  sides. 
Jfonagon  has  9  sides. 
Decagon  has  10  sides. 


710.  A  Perimeter  is  the  boundary-line  of  a  plane  figure. 

711.  The  Altitude  of  a  figure  is  the  line  which,  being  drawn 
perpendicular  to  the  base,  measures  the  height  of  the  figure. 

712.  By  Geometry  may  be  established  the  following — 

713.  Principles, — 1.    The  area  (Art.  395)  of  a  parallelo- 
gram is  equal  to  the  product  of  the  base 
by  the  altitude. 


That  is,  if  th«  altitude  of  a  parallelogram 
be  4  inches,  and  the  base  be  8  inches,  the 

area  will  be  as  many  square  inches  as  the  product  of  8  by  4,  or  32  square 
inches. 

2.  Tlie  area  of  a  trapezoid  is  equal  to  the  product  of  half  of 

the  sum  of  the  2^(i^<^ll^l  sides  by  the  

"Hitude.  A  \ 

That  is,  if  the  altitude  of  a  trapezoid  be      /     I \ 

4  inches,  the  longer  of  the  parallel  sides  be 

14  inches,  and  the  shorter  of  the  parallel  sides  10  inches,  the  area  -will  be 

as  many  square  inches  as  the  product  of  ^^X^^  by  4,  or  48  square  inches. 

3.  The  area  of  a  triangle  is  equal  to  the  product  of  half  the 
base  by  the  altitude. 

That  is,  if  the  altitude  of  a  triangle  be  4  inches, 
and  the  base  8  inches,  the  area  will  be  as  many 
square  inches  as  the  product  of  ?  by  4,  or  16 
square  inches. 


MENSURATION. 


281 


4.   The  area  of  a  trapezium,  or  of  any  polygon,  is  equal  to  the 
su7n  of  the  areas  of  any  set  of  triangles  into 
which  it  may  be  resolved. 

That  is,  if  tlie  trapezium  be  separated  into  two  tri- 
angles, the  sum  of  the  areas  of  these  triangles  will  be 
the  area  of  the  trapezium. 

Note. — The  area  of  a  regular  polygon  is  equal  to  the  square  of  its  side  multiplied  by 
tlie  area  staudiug  against  its  name  in  the  following  table: 


Name. 

Area. 

Name. 

Area. 

Name. 

Area. 

Triangle, 
Pentagon, 
Hexagon, 

.433013 
1.720477 
2.598076 

Heptagon, 

Octagon, 

Nonagon, 

3.633913 

4.828427 
6.181824 

Decagon, 

Undecagon, 

Dodecagon, 

7.694209 

9.36564 

11.196152 

Also,  the  area  of  any  triangle  is  equal  to  the  square  root  of  the  product 
of  half  the  sum  of  the  three  sides  by  the  three  remainders  found  by  sub- 
tracting each  side  separately  from  half  the  sum  of  the  three  sides. 


PROBJ.EMS. 

1.  What  is  the  area  of  a  right-angled  triangle  whose  base  is 
340  feet,  and  perpendicular  120  feet? 

Ans.  20400  square  feet. 

2.  How  many  acres  in  a  triangular  lot  whose  base  is  517 
yards,  and  perpendicular  341  yards  ? 

Ans.  18  acres  34  square  rods. 

3.  Required  the  area  of  a  parallelogram  whose  length  is  36 
feet,  and  altitude  15  feet.  Ans.  540  square  feet. 

4.  The  parallel  sides  of  a  field  in  the  form  of  a  trapezoid 
are  8  chains  15  links,  and  10  chains  45  links,  and  the  perpen- 
dicular distance  between  the  parallel  sides  is  6  chains  24  links. 
What  is  its  area  in  acres  ? 

Ans.  5  acres  128.512  square  rods. 

5.  How  many  square  yards  in  a  garden  in  the  form  of  a 
trapezium  whose  diagonal  is  20  yards,  and  the  perpendiculars 
on  it  to  opposite  angles  are  4.2  yards  and  3.8  yards  ? 

Ans.  80  square  yards. 

24* 


282 


MENS  URA  TION. 


6.  What  is  the  area  of  a  triaiigidar  field  whose  sides  are 
2569,  5025  and  4900  links?    Ans.  61  acres  79.68  square  rods. 

7.  How  many  square  yards  of  surface  in  the  gable  of  a 
house  whose  breadth  is  20  feet  6  inches,  and  perpendicular 
height  10  feet  4  inches? 

Ans.    11  square  yards  Q\^  square  feet. 

8.  How  many  square  yards  of  painting  are  there  in  the 
ceiling  of  a  room  whose  length  is  24  feet  and  breadth  15  feet 
6  inches  ?  Ans.  41  sq.  yd.  3  sq.  ft. 

9.  How  many  acres  are  contained  in  a  farm  of  the  form  of 
a  regular  decagon  whose  side  is  2050  links  ? 

Ans.  323  acres  55.86  square  rods. 


EIGHT-ANGLED  TRIANGLES. 
714.  By  Geometry  may  be  established  the  following 

Principles. — 1.  The  square  of 
the  hypotenuse  is  equal  to  the  swn 
of  the  squares  of  the  other  tivo  sides. 

That  is,  if  the  base  be  4  and  the  per- 
pendicular 3,  the  square  of  the  hypote- 
nuse will  equal  4^  +  3'^  or  16  +  9, 
and  the  hypotenuse  itself  will  equal 
l/16  +  9  =  5. 

2.    The   square   of   either   of   the 
sides  forming  the  right  angle  of  the 
triangle  is  equal  to  the  square  of  the  hypotenuse  minus  the  square 
of  the  other  side. 

That  is,  if  the  base  be  4  and  the  hypotenuse  be  5,  the  square  of  the 
perpendictilar  will  equal  5^  —  4^,  or  25  —  16;  and  the  perpendicular 
will  equal  |/25  —  16  =  3.  Also,  if  the  perpendicular  be  3  and  the 
hypotenuse  be  5,  the  square  of  the  base  will  equal  b'^  —  3'^,  or  25  —  9, 
and  the  base  itself  will  equal  |/25  —  9  =  4. 

Note. — For  practical  purposes,  such  as  squaring  the  foundations  of  a  building,  a  right- 
angled  triangle  is  readily  constructed  by  so  fastening  two  |)ieci's  of  wood  togithcr,  ono 
8  fi'et  long  and  the  other  0  feet  lung,  that  the  distanco  between  the  outer  sides  of  tho  two 
ends  shall  be  10  foot. 


MENSURATION.  283 

PROBIjEMS. 

1.  The  base  of  a  right-angled  triangle  is  85  yards,  and  the 
perpendicular  132  yards.     Find  the  hypotenuse. 

Solution. 

85^=7225,  and  132^=17424;  7225  + 17424-=24640  ; 
and  V 24649  =  157,  Ans. 

2.  A  ladder  40  feet  long  standing  upon  a  level  street  reaches 
to  a  window  in  a  building  32  feet  from  the  ground.  Required 
the  distance  of  the  foot  of  the  ladder  from  the  building. 

3.  A  carpenter  building  a  house  32  feet  Avide  wishes  to  have 
the  gable  end  16  feet  high.  How  long  must  the  raftei-s  be 
made  ? 

4.  A  park  whose  length  is  105  rods  and  width  88  rods  has 
a  walk  running  diagonally  across  it.  What  is  the  length  of 
the  walk?  ^n.s.  137  rods. 

5.  At  $1.10  per  rod,  what  will  be  the  cost  of  fencing  a  lot, 
in  the  form  of  a  right-angled  triangle,  whose  sides  forming  the 
right  angle  are  264  rods  and  23  rods? 

6.  A  boy  flying  his  kite  found  that  he  had  let  out  210  feet 
of  the  string  when  the  kite  lodged  upon  the  top  of  a  flag- 
staff whose  height  was  91.53  feet.  Required  the  distance  of 
the  boy  from  the  foot  of  the  stafi*.  Aiis.  189  feet. 

CIECLES. 
715.  By  Geometry  may  be  established  the  following 

Principles. — 1.  The  ratio  of  the  circum- 
ference to  the  diameter  is  expressed  approx- 
imately by  3^,  or  by  3.1416 ;  and  of  the 
diameter  to  the  circumference, by  -^-o,  or. 3183. 

2.  The  area  of  a  circle  is  equal  to  the 
product  of  half  the  circumference  by  half  the 
diameter. 

3.  The  area  of  a  circle  is  also  equal  to  the  2}roduet  of  the 
square  of  the  diameter  by  .7854,  or  to  the  product  of  the  square 
of  the  circuvference  by  .07958. 


284 


MENSURATION. 


4.  The  side  of  the  largest  square  that  can  he  inscribed  in  a 
given  circle  is  equal  to  the  square  root  of  half  the  square  of  the 
diameter,  or  to  .7071  times  the  diameter. 

I'll  OBLliMS. 

1.  The  diameter  of  a  circular  flower-plat  is  17.5  feet. 
What  is  its  circumference  ?  Ans.  55  feet. 

2.  The  circumference  of  a  circular  pond  is  506  yards. 
How  many  feet  is  its  diameter?  Ans.  483  feet. 

3.  How  many  acres  in  a  circular  lot  whose  diameter  is  476 
links?  Ans.  1  acre  124|||  square  rods. 

4.  What  is  the  side  of  the  largest  square  that  can  be  laid 

out  in  a  circular  enclosure  whose  diameter  is  8  rods  ? 

Ans.  5.6568  rods. 
VOLUMES. 

716.  A  Prism  is  a  volume  whose  ends  are  equal  and  parallel 
polygons,  and  whose  sides  are  parallelograms. 

A  prism  is  triangular,  rectangular,  etc.,  according  as  its  ends 
are  triangles,  rectangles  (Arts.  313,  693),  etc. 

717.  A  Cylinder  is  a  volume  of  uniform  diameter,  bounded 
by  a  curved  surface  and  two  equal  and  parallel  circles. 


A  Rectangular  Prism.  Ctlindek. 

718.  A  Pyramid  is  a  volume  Avhose  base  is  a  polygon  and 
whose  sides  are  triangles  meeting  in  a  point,  called  the  vertex. 

A  pyramid  is  triangular,  quadrangular,  etc.,  according  as  its 
base  is  a  triangle,  quadrilateral,  etc.,  and  its  taper  surface  is 
that  of  the  sum  of  its  triangular  faces. 

719.  A  Cone  is  a  volume  Avhose  base  is  a  circle,  from  which 
the  remaining  surface  tapers  uniformly  to  a  point  or  vertex. 


MENS  UEA  Tl  ON. 


285 


720.  The  Slant  Height  of  a  pyramid  or  cone  is  the  shortest 
line  that  can  be  drawn  from  the  vertex  to  the  perimeter  or 
circumference  of  the  base. 


A  Pyramid. 


A  Cone. 


The  curved  surface  of  a  cone  is  equal  to  the  surface  of  a 
triangle  whose  base  is  the  circumference  of  the  cone  and 
whose  altitude  is  the  slant  heisrht. 


721.  A  Frustum  of  a  pyramid  or  cone 
is  that  which  remains  after  cutting  off  the 
upper  part  of  it  by  a  plane  parallel  to  the 
base. 


722.  A  Sphere  is  a  volume  bounded  by  a 
surface,  all  points  of  w^hich  are  equally  dis- 
tant from  a  point  within,  called  the  center. 


A  Spbebe. 


723.  By  Geometry  may  be  established  the  following 

724.  Principles. — 1.  The  cubic  contents  of  a  prism  or  cyl- 
inder are  equal  to  the  product  of  the  area  of  the  base  by  the 
altitude. 

2.  The  cubic  contents  of  a  pyraviid  or  cone  are  equal  to  one 
third  of  the  product  of  the  area  of  the  hose  by  the  altitude. 

3.  Tlie  cubic  contents  of  a  frustum  of  a  pyraviid  or  cone 
equal  the  product  of  the  sum  of  the  areas  of  the  two  ends  and  the 
square  root  of  their  product, by  one  third  of  the  altitude. 


286  MENSURATION. 

4.  The  surface  of  a  sphere  is  equal  to  the  product  of  the  cir- 
cumference by  the  diameter,  or  to  the  j^roduct  of  the  square  of  the 
diameter  by  3.1416. 

5.  The  cubic  contents  of  a  sphere  are  equal  to  the  product  of 
the  surface  by  one  sixth  of  the  diameter,  or  to  the  product  of  the 
cube  of  the  diameter  by  .5236. 


FROBZEMS. 

1.  Kequired  the  cubic  contents  of  a  squared  beam  whose 
length  is  32  feet,  and  whose  breadth  and  thickness  are  each  1 
foot  6  inches.  ^?««-  72  cubic  feet. 

2.  A  pentagonal  stone,  whose  side  is  2  feet,  has  an  altitude 
of  6  feet.     What  are  its  cubic  contents  ? 

Am.  41.291448  cubic  feet. 

3.  The  circumference  of  a  cylindrical  log  is  5  feet  6  inches, 
and  its  length  20  feet.     What  are  its  cubic  contents  ? 

Ans.  48.1458  cubic  feet. 

4.  The  sides  of  the  base  of  a  triangular  pyramid  are  11, 13, 
20  feet,  and  the  altitude  is  36  feet.  What  are  the  cubic  con- 
tents? -4ns.  792  cubic  feet. 

5.  What  are  the  cubic  contents  of  a  conical  tower  whose 
circumference  is  13.20  yards,  and  altitude  27  yards? 

Ans.  124.74  cubic  yards. 

6.  What  is  the  curved  surface  of  a  cone  whose  circumference 
is  9.4248  yards,  and  slant  height  15  yards? 

7.  What  is  the  weight  of  a  marble  monument  in  the  form 
of  a  square  pyramid,  each  side  of  the  base  being  3  feet  and 
its  altitude  10  feet,  at  166  pounds  per  cubic  foot? 

Ans.  4980  pounds. 

8.  What  will  it  cost  to  paint  a  spire,  in  the  form  of  a  quad- 
rangular pyramid,  whose  slant  height  is  80  feet,  and  each  side 
of  the  base  18  feet,  at  $.50  per  square  yard  ?  Ans.  $160. 

9.  Required  the  surface  in  square  inches  of  a  sphere  whose 
diameter  is  4  feet  8  inches.  Ans.  9856  square  inches. 

10.  Each  side  of  the  greater  end  of  a  piece  of  squared  tim- 
ber is  28  inches,  each  side  of  the  lesser  end  14  inches,  and  its 


MENSURATION.  287 

length  18  feet  9  inches.     How  many  cubic  feet  does  it  con- 
tain? Ans.  59.5486+  cubic  feet. 

11.  What  are  the  cubic  contents  of  a  sphere  whose  diameter 
is  10  feet  6  inches  ?  Ans.  606.375  cubic  feet. 

12.  How  many  gallons  of  water  will  a  boiler  hold  whose 
shape  is  that  of  a  half  sphere  25  inches  in  diameter  ? 

Ans.  17.7154+  gallons. 

SIMILAR  FIGURES. 

725.  Similar  Figures  are  such  as  are  of  the  same  form,  and 
differ  from  each  other  only  in  size. 

726.  Similar  figures  have  such  a  relation  to  each  other  that — 

1.  The  like  dimensions  of  similar  figures  are  pi'oportional. 

2.  The  areas  of  similar  figures  are  to  each  other  as  the  squares 
of  their  corresponding  dimensions. 

3.  The  cubic  contents  of  similar  volumes  are  to  each  other  as 
the  cubes  of  their  corresponding  dimensions. 

PltOBZiESIS. 

1.  The  sides  of  a  triangular  lot  are  15,  28  and  41  rods,  and 
the  area  is  126  square  rods.  What  are  the  sides  of  a  similar 
lot  whose  area  is  56  square  rods?   . 

Solution. 

By  Art.  726—2,  126  \  56  \\  W  :  100. 
Hence,  the  corresponding  sides  of  the  similar  lot, 

\/ 100  =  10,  the  one  side. 

Then  by  Art.  726—1,  15:  10:  \  28  \  18j, 

and  15:  10::  41: 27 j-  , 

2.  The  area  of  a  triangle  whose  base  is  17  chains  is  3  acres 
71  square  rods.  What  is  the  area  of  a  similar  triangle  whose 
base  is  57  chains?  Ans.  38  acres  114.46  square  rods. 

3.  If  the  side  of  a  square  field  containing  10  acres  is  40 
rods,  what  must  be  the  side  of  a  similar  field  containing  one 
fourth  as  many  acres  ?  Aivs.  20  roda 


288  MEXSUEATION. 

4.  The  area  of  a  circle  whose  diameter  is  100  feet,  is  7854 
square  feet;  what  is  the  area  of  a  circle  whose  diameter  is  10 
feet? 

5.  What  are  the  dimensions  of  a  rectangular  garden  contain- 
ing 180  square  rods,  whose  length  is  to  its  breadth  as  5  to  4  ? 

Solution. 

By  Art.  398,  5x4  =  20,  the  area  of  a  rectangle  5  by  4. 
Then,      20  \  180  \\  B""  \  225 ; 
and,        y'225  =  15,  the  greater  side. 
Also,       5  :  4::  15  :  12,  the  mialler  side. 

6.  The  three  edges  of  a  rectangular  volume  are  3,  4,  7 
inches ;  what  are  sides  of  a  similar  volume  whose  contents  are 
777924  cubic  inches  ?  Aiis.  63,  84  and  147  inches. 

.7.  If  a  ball  3  inches  in  diameter  weigh  4  pounds,  how  much 
will  a  ball  of  like  density  weigh  whose  diameter  is  6  inches? 

Ans.  32  pounds. 

8.  What  are  the  dimensions  of  a  rectangular  field  contain- 
ing 30  acres,  whose  length  is  to  its  breadth  as  4  to  3  ? 

9.  A  stack  of  hay  5  feet  high  Aveighs  100  pounds ;  how  high 
must  be  a  similar  stack  to  weigh  3  tons  400  pounds  ? 

10.  If  the  edge  of  a  cube  of  Quincy  granite  weighing  165.75 
pounds  is  12  inches,  what  is  the  edge  of  a  cube  weighing  500 
pounds  ? 

11.  If  a  ball  of  thread  4  inches  in  diameter  should  be  re- 
duced to  half  that  diameter,  what  part  of  the  thread  will  re- 
main ?  Am,  \. 


TEST    QUESTIONS. 

■ff 

727.— 1.  Of  what  does  Mensuration  treat?  What  is  a  point? 
What  is  a  line?  (Art.  301.)     A  curved  line ?     What  are  parallel  lines ? 

2.  What  is  an  Angle?  (Art.  309.)  How  is  a  right  angle  formed? 
What  name  is  given  to  an  angle  less  than  a  right  angle?  To  an  angle 
greater  than  a  right  angle  ? 

3.  What  is  a  Surface?  (Art.  .302.)  What  is  a  plane?  The  area  of  a 
fignre  ?  The  base  of  a  figure  ?  The  altitude  of  a  figure  ?  The  perim- 
eter?    A  diagonal? 


MI:^'S  URA  TION.  289 

4.  What  is  a  Triangle?  An  equilateral  triangle?  An  isosceles 
triangle?  A  scalene  triangle?  A  right-angled  triangle?  An  acntc- 
angled  triangle?  An  obtuse-angled  triangle?  How  is  the  area  of  a  tri- 
angle found  ? 

5.  What  is  a  Quadrilateral?  A  square?  (Art.  312.)  A  rectangle? 
(Art.  313.)  A  parallelogram?  When  is  a  })arallelogram  a  rhomboid? 
When  a  rhombus?  When  is  a  quadrilateral  a  trapezoid  ?  When  a  tra- 
pezium ? 

6.  How  is  the  Area  of  a  rectangle  found  ?  (Art.  398.)  How  is  the 
area  of  a  parallelogram  found  ?  How  is  the  area  of  a  trapezoid  found  ? 
How  is  the  area  of  a  trapezium  found  ? 

7.  What  is  a  Polygon  ?  A  regular  polygon  ?  How  many  sides  has 
a  pentagon  ?  A  hexagon  ?  A  heptagon  ?  An  octagon  ?  A  nonagon  ? 
A  decagon  ?     How  is  the  area  of  a  polygon  found  ? 

8.  What  is  the  longest  side  of  a  right-angled  triangle  called?  To 
what  is  the  square  of  the  hypotenuse  equal?  When  any  two  sides  of  a 
right-angled  triangle  are  given,  how  is  the  other  side  found  ? 

9.  What  is  a  Circle?  The  diameter  of  a  circle?  (Art.  347.)  The 
circumference  of  a  circle?  (Art.  345.)  What  is  the  ratio  of  the  circum- 
ference to  the  diameter?  Of  the  diameter  to  the  circumference?  How 
is  the  area  of  a  circle  found  ? 

10.  What  is  a  Volume?  (Art.  303.)  What  is  a  cube?  (Art.  317.) 
A  rectangular  volume  ?  What  are  the  cubic  contents  of  a  Vv>lume  ?  How 
are  the  cubic  contents  of  a  rectangular  volume  found?   (Art.  405.) 

11.  What  is  a  Prism  ?  A  cylinder?  How  t>r^  the  cubic  contents  of  a 
prism  or  of  a  cylinder  found  ? 

12.  What  is  a  Pyramid  ?  A  cone  ?  The  slant  height  of  a  pyramid 
or  a  cone  ?  A  frustum  of  a  pyramid  or  cone  ?  How  are  the  cubic  con- 
teuts  of  a  pyramid  or  a  cone  found  ?  The  cubic  contents  of  a  frustum 
of  a  pyramid  or  a  cone? 

13.  What  is  a  Sphere  ?  To  what  is  the  surface  of  a  sphere  equal  ? 
To  what  are.  the  cubic  contents  equal  ? 

14.  What  are  Similar  Figures  ?     What  relations  have  like  dimen- 
sions of  similar  figures?     The  areas  of  similar   figures?      The  cubic* 
contents  of  similar  figures? 

15.  How  does  a  curved  line  differ  from  a  straight  line?  An  acute 
angle  from  an  obtuse  angle  ?  What  is  the  perimeter  of  a  triangle  ?  Of 
a  square  ?     Of  a  circle  ? 

16.  What  kind  of  a  prism  is  a  rectangular  solid  ?  What  kind  of  a 
pyramid  is  one  having  a  four-sided  base?  How  does  a  cone  difl'er  from 
a  pyramid  ?     How  does  a  sphere  differ  from  a  cylinder? 

2J 


290 


MEASUREMENTS  OF  BOARDS  AND    TIMBER. 


SECTION   LXVIII. 
MEASUREMEJfTS  OF  BOARDS  AJ{I)  TIMBER. 

728.  A  Board  Foot  is  1  foot  long,  1  foot  broad  and  1  inch 
thick;  hence, 

12  hoard  feet  are  1  cubic  foot. 


729.  Lumber,  or  sawed  timber, 
is  estimated  in  board  feet. 
^^■^"^^B^^^MM^rfy^^r  730,  Eoiiml  Timber  is  usually 

estimated  in  cubic  feet. 

731.  Squared  or  Hewn  Timber  is  estimated  either  in  board 
feet  or  cubic  feet. 

732.  The  Mean  Girt  of  a  tapering  log  is  the  circumference, 
clear  of  bark,  one-third  the  distance  from  the  larger  to  the 
smaller  end  of  the  log. 

733.  The  Mean  Breadth  and  Thickness  of  tapering  squared 
timber  is  the  breadth  and  thickness  of  the  timber  measured  at 
the  middle  of  its  leno-th. 


CASK    I. 

Dimensions  of  Lumber,  or  Squared  Timber,  giren,  to  And  the 
Contents, 

734. — Ex.  1.  AVhat  is  the  number  of  board  feet  in  a  board 
24  feet  long,  18  inches  wide  and  1  inch  thick  ? 

Solution.— 18  inches  -  ]  \  feet.     24  X  if  X  1  =  ^^2^^  ==  ^^-  ^^^^ 
number  of  board  feet  required. 


MEASUREMENTS    OF  BOARDS  AND    TIMBER.  291 

2.  How  many  cubic  feet  in  a  piece  of  hewn  timber  45  feet 
long,  whose  breadth  and  thickness  are  15  and  14  inches  ? 

Solution. — 15  and  14  inches  =  {|  and  if  feet.     45  X  B  X  11  = 


45  X  15X14 


65.625  cubic  feet. 


144 

8.  What  are  the  contents  in  board  feet  of  a  joist  IG  feet 
long,  4  inches  thick,  and  tapering  in  breadth  from  6  inches  to 
4  inches?  Ans.  26|. 

The  mean  of  the  tapering  breadth  is  5  inches,  which  is  used  in  the 
computation. 

735.  Rule   for   Finding   the    Contents  of  Lumber    and   Squared 

Tmber.—Multiplj/  the  length  in  feet  by  the  hreadth 
and  thichness  in  inches,  and  divide  by  12  for  board 
feet,  or  by  IJfJi-  for  cubic  feet. 

If  the  hunber  or  squared  timber  tapers,  the  mean  breadth  and  thick- 
ness must  be  used  in  the  computation. 

A  common  rule  for  estimating  tapering  squared  timber  is — 

Add  together  the  areas  of  the  two  ends  in  square 
inches,  and  multiply  half  the  suin  by  the  length  in 
feet,  and  divide  by  12  for  board  feet,  or  by  IJfJf  for 
cubic  feet. 

pitoBLJi:3i:s. 

1.  What  are  the  contents  of  a  plank  whose  length  is  20 
feet,  breadth  16  inches  and  thickness  3  inches? 

Ans.  80  board  feet. 

2.  A  beam  is  30|-  feet  long,  22  inches  broad  and  15  inches 
thick.     What  are  its  contents  ?  Ans.  843^  board  feet. 

3.  The  length  of  a  piece  of  timber  is  9.8  feet,  and  its  mean 
breadth  and  thickness  2.6  and  1.5  feet.  What  are  the  cubic 
contents  ?  Ans.  38.22  cubic  feet. 

4.  Bought  20  joists,  each  18  feet  long,  5  inches  wide  and  3 
inches  thick,  at  S30  a  thousand  feet,  board  measure.  What  did 
they  cost  me?  ,  ^ns.  813.50. 

5.  The  breadth  and  thickness  of  one  end  of  a  stick  of 
timber,  whose  length  is  17  feet  3  inches,  are  36  and  20  inches, 


292  MEASUREMENTS  OF  BOARDS  AND   TIMBER. 

and  of  tlie  otlier  end  18  and  10  inches.  What  are  its  cubic 
contents  by  the  rule,  allowing  the  mean  breadth  and  thickness 
to  be  27  and  15  inches ;  and  what  the  true  contents,  measured 
as  a  frustum  of  a  pyramid  ? 

Ans.  By  the  rule,  48,5156  +  cubic  feet ;  true  con- 
tents, 50.3125  cubic  feet. 

C^SE    II. 

Lengdi  and  Mean  Girt  of  Round  Timber  g-iren,  to  find  tlie  Contents. 

736. — Ex.  1.  What  are  the  cubic  contents  of  a  piece  of 
round  timber  Avhose  mean  girt  is  100  inches  and  length  18 
feet? 

Solution.— 1^  of  mean  girt  =  \  of  100  =  25;  square  of  \  of  mean 
cfirt  =  252  =  625 ;  625X18  =  11250;  and  11250 -^- 144  =  78.125,  the 
nnmlier  of  cubic  feet  ref|nired. 

2.  The  mean  girt  of  a  log  is  88  inches,  and  the  length  of 
the  log  is  40  feet.     What  are  the  cubic  contents  ? 

Avs.  134.44  cubic  feet. 

737.  Rule  for   Finding  the  Cubic  Contents  of  Round  Timber.— 

Multiply  the  square  of  one  fourth  of  the  mean  girt  in 
inches  by  the  length  iit  feet,  and  divide  by  1^^. 

Note. — The  rule  gives  about  one  fifth  less  than  the  exact  quantity,  one  fifth  heing 
til  lowed  for  crooks  and  waste  in  working. 

The  exact  cubic  contents  may  be  found  very  nearly  by  multiplying  the  square  of  ane 
Jiflh  of  tlie  mean  girt  in  feel  by  twice  the  length  in  feet. 

PROJil.  EMS. 

1.  The  length  of  a  log  is  32  feet  6  inches,  and  its  mean 
girt,  after  allowing  for  the  bark,  is  60  inches.  What  are  the 
contents  by  the  rule,  and  what  by  the  note  under  the  rule  ? 

Ans.  By  the  rule,   50.78125  cubic   feet;    by  the 
note,  65  cubic  feet. 

2.  What  is  the  value  of  a  pine  log  30  feet  long,  and  whose 
mean  girt  is  10  feet,  at  $20  per  ton  of  40  cubic  feet? 

Ans.  S93.75. 

3.  The  circunifcrcncc  of  a  piece  of  round  timber  is  6  feet  8 
inches,  and  its  length  24  feet.  What  are  its  contents  by  the 
rule,  and  what  as  a  cylinder?  (Art.  724 — 1.) 


MEASUREMENTS   OF  STONE,  AND  BRICK-  WORK.        293 


SECTION    LXIX. 

MEASVREMEKTU  OF  STOKE,  AKD  BRICK- 
WORK. 

738.  Stone  Masonry  is  usually  estimated  by  the  cubic  foot 
or  by  the  perch. 

739.  Brick-Layiiiisr  is  generally  estimated  by  the  thousand 
bricks. 

740.  A  Perch  of  stone-Avork  is  16^-  feet  long,  1  foot  deep 
and  1^  feet  thick,  and  is  equivalent  to  24|  cubic  feet. 

741.  Bricks  are  of  various  dimensions. 

Philadelphia  or  Baltimore  front  bricks  are  8|,  4|  and  2|  inches ; 
North  River  bricks,  8,  ?>\  and  2|" inches;  Maine  bricks,  7|,  3|  and  2f  ; 
and  Milwaukee  bricks,  8|,  4|  and  2|-  inches. 

C^SE    I. 

Dimensions  of  Stone-ivork  j^iren,  to  And  the  Number  of  Perclies. 

742. — Ex.  1.  How  many  perches  of  stone-work  in  a  M'all 
66  feet  long,  4  feet  high  and  3  feet  wide  ? 

Solution. 
66y.4'Xo=792,  mimher  of  cubic  feet. 
792  -^  2^.75  =  32,  number  of  j^erches. 

743.  Rule  for  Finding  the  Number  of  Perciies  of  Stone-work.— 

Find  tlie  contents  in  cubic  feet,  and  divide  hy  2Jj..7o. 

PJtOBI.EMS. 

1.  What  are  the  contents  in  perches  of  a  stone  wall  whose 
dimensions  are  24  feet  3  inches,  10  feet  9  inches  and  2  feet? 

Ans.  21.065  +  perches. 

2.  What  will  it  cost,  at  83.25  per  perch,  for  the  stone-  and 
mason-work  of  a  cellar  8  feet  deep,  under  a  house  whose 
length  and  width  are  411  ^nd  33  feet,  the  wall  of  the  cellar  to 
be  1\  feet  thick,  and  no  allowance  to  be  made  for  corners  or 
openings?  Ans.  $234. 

25  » 


294       MEASUREMEl^TS   OF  STONE,   AND  BRICK -WORK. 

ca-sk;  II. 

Dimensions  of  Bricks  and  Thickness  of  Mortar  of  Brick-ivork 
giyen,  to  find  the  ^"nmber  of  Bricks. 

744. — Ex.  1.  The  width  of  a  wall  is  10^  inches,  laid  of 
Maine  bricks,  in  courses  of  mortar  \  of  an  inch  thick.  How 
many  bricks  has  it  in  a  cubic  foot  ? 

Solution. 
7.5  +  (.25  X2)^2-=  7.75,     length  of  brick  and  joint. 
2.S75  +(.25X2)^2  =  2. 625,  width  of  brick  and  joint. 

7.75  X  2. 625  =  20.34375,  area  of  face. 
10.5^3  =  3.5  ;  20.34375  X  3.5  =  71.2  +  cubic  inches. 
1728  -  71.2  =  24.269  -^,  the  number  of  bricks. 

745.  Rule  for  Finding  tiie  Number  of  Bricl<s  in   Brick-work.— 

To  the  face  dimensions  of  the  hind  of  bricks  used, 
add  half  the  thichness  of  the  mortar  in  which  they 
are  laid,  and  find  the  area.  Multiply  this  area  by 
the  quotient  obtained  by  dividing  the  width  of  the 
wall  by  the  member  of  briclcs  of  which  it  is  composed, 
and  the  product  ivill  be  the  contents  in  cubic  inches. 
Divide  1728  cubic  inches  by  these  contents,  and  the 
quotient  will  be  the  number  of  briclcs  in  a  cubic  foot. 

PBOBLEJtrS. 

1.  A  wall  of  Milwaukee  bricks  is  40  feet  long,  7  feet  high 
and  21f  inches  thickj  and  the  courses  of  mortar  in  which  the 
bricks  are  laid  are  \  of  an  inch  thick.  How  many  thousand 
bricks  are  there  in  the  wall  ? 

2.  The  width  of  a  wall  is  llf  inches,  laid  of  North  River 
bricks,  in  courses  of  mortar  \  of  an  inch  thick.  How  many 
bricks  has  it  in  a  cubic  foot? 

8  A  wall  of  Philadelphia  bricks  is  20  feet  long,  5  feet  high 
and  12f  inches  thick,  and  the  courses  of  mortar  in  which  the 
bricks  are  laid  are  \  of  an  inch  thick.  Wliat  must  have  been 
the  cost  of  the  bricks  at  $12  per  thousand?        Atis.  $28.23. 


MEASUREMENTS   OF  GRAIN  AND  HAY.  295 

SECTION    LXX. 
MEASUREMENTS   OF  GBAI:N'  AjYD  HAY. 

746.  (irain  is  usually  estimated  in  this  country  by  the  bushel 
or  by  the  cental.    (Art.  332.) 

747.  The  Standard  Busliel  in  the  United  States  contains 
2150.4  cubic  inches.    Hence,  a  cubic  foot  is  nearly  .8  of  a  bushel. 

748.  Hay  is  usually  bought  and  sold  by  the  ton. 

About  550  cubic  feet  of  clover,  or  450  feet  of  meadow-hay,  well  settled, 
as  an  average,  in  large  mows,  make  a  ton. 

749.  Rules  for  Estimating  Grain.— i.  To  find  tlio  qiinntUy, 
in  bushels,  of  graiit  in  a  hin  or  ivagon,  multiply  the 
contents  in  cubic  feet  by  .8. 

'2.  To  find  the  quantity,  in  bushels,  of  grain  ivhen 
heaped  upon  a  floor,  make  the  heap  in  the  form  of  a 
cone,  and  multiply  tJie  area  of  the  base  by  one-third 
of  the  altitude,  and'  the  result  by  .8. 

If  the  grain  be  heaped  against  the  side  of  a  wall  in  the  form  of  a  half 
cone,  take  half  the  result  by  Rule  2,  or,  if  heaped  against  an  inner 
corner,  take  one  fourth. 

750.  Rule  for  Estimating  Hay.— i.  To  find  tlie  quantity ,  in 
tons,  of  hay,  in  mows,  well  settled,  divide  the  contents 
by  ooO  for  clover,  or  by  Jf50  for  meadow-hay. 

1.  How  many  bushels  of  wheat  will  a  bin  hold  that  is  0 
feet  long,  4  feet  wide  and  5  feet  deep  ?  Ans.  96. 

2.  A  wagon  9  feet  long,  31  feet  wide  and  3  feet  deep  is  two 
thirds  full  of  shelled  corn.  How  many  bushels  does  it 
contain  ? 

8.  A  farmer  had  a  heap  of  oats,  which  when  made  in  a 
conical  form,  measured  in  the  circumference  of  the  base  22 


2i)6  GA  UGING. 

feet,  and  in  altitude  6  feet.      How  many  bushels  does  it  con- 
tain ?  Ans.  61.6. 

4.  In  the  inner  corner  of  a  building  corn  is  heaped  in  the 
form  of  a  quarter  cone,  whose  altitude  is  6  feet  and  slant 
height  10  feet.     What  is  the  quantity  of  corn? 

Ans.  80|^f  bushels. 

5.  A  heap  of  grain,  piled  against  a  wall,  measures  a  height 
of  9  feet  and  a  semi-circumference  of  33  feet.  How  many 
bushels  does  it  contain  ?  A71S.  415.8. 

6.  A  bin  6  feet  long,  4  feet  Avide  and  5  feet  deep  is  filled 
with  rye.     What  is  its  value  at  S2  per  cental  ? 

Ans.    $107.52. 

7.  How  many  tons  of  meadow-hay  in  a  portion  of  a  stack, 
dry  and  settled,  which  is  10  feet  long,  9  feet  wide  and  6  feet 
high  ? 

8.  What  is  the  value,  at  $30  a  ton,  of  a  mow  of  clover- 
hay  that  is  24  feet  long,  12  feet  wide  and  9  feet  high  ? 


SECTION    LXXI. 
GA  UGIA^G. 

751.  Gauging'  is  the  process  of  finding  the  capacity  of  casks 

and  other  vessels. 

752.  The  Mean  Diameter  of  a  cask  is  the  diameter  of  an 
equivalent  cylinder  having  the  same  length  as  the  cask. 

It  is  nearly  equal  to  the  head  diameter  plus  two  tliirds  of  tlie  differ- 
ence between  that  and  the  lonjj  diameter,  or  three  fifths  v>iien  the  staves 
are  but  slightly  curved. 

753.  The  Ullage,  or  wantage,  of  a  cask  is  the  (quantity  it 
lacks  of  being  full. 

754.  Rules  for  Gauging.— i.  Multiply  the  product  of  the 
square  of  the  mean  diameter  and  the  length  or  depth, 
of  the  cash,  expressed  in  inches,  hy  .003 Jf,  and  the 
result  jvill  be  its  capacity  in  gallons. 


GA  UGING.  297 

2.  Multiply  the  square  of  one  third  of  the  sum  of 
the  head,  ineaiv  aiul  hung  diaiueters ,  expressed  in 
inches,  by  the  height  of  the  liquid  in  inches,  and  that 
product  by  .003 Jj.,  and  the  result  will  he  the  contents 
of  ayv  ullage  cash. 

PROBLEMS. 

1 .  How  many  gallons  is  the  capacity  of  a  cask  whose  length 
is  40  inches  and  mean  diameter  25  inches?     Ans.  85  gallons. 

2.  Required  the  quantity  of  vinegar  in  a  cask  whose  bung 
and  head  diameters  are  37  and  28  inches,  and  the  height  of 
the  liquid  10  inches.  Ans.  37.026  gallons. 

3.  How  much  Avill  a  cask  of  molasses  cost  whose  mean 
diameter  is  30  inches  and  length  36  inches,  at  'S.55  per  gallon  ? 

Ans.  $60,083%. 


TEST    QUESTIONS. 

755. — 1.  What  are  the  dimensions  of  a  Board  Foot?  How  many 
board  feet  are  one  cubic  foot  ? 

2.  What  kind  of  timber  is  Lumber?  In  what  is  squared  or  hewn 
timber  estimated  ?  What  are  the  rules  for  finding  the  contents  of  lumber 
and  squared  timber  ? 

3.  In  what  is  RocND  Timber  estimated  ?  What  is  the  mean  girt  of  a 
tapering  log?  What  is  the  mean  breadth  and  thickness  of  tapering 
squared  timber  ?  What  is  the  rule  for  finding  the  cubic  contents  of  round 
timber? 

4.  By  what  is  vStoxe  Masoxry  estimated  ?  What  is  a  perch  of  stone- 
work?  What  is  the  rule  for  finding  the  number  of  perches  of  stone-work  ? 

5.  IIow  is  Bricklaying  usually  estimated  ?  What  is  the  rule  for 
finding  the  number  of  bricks  in  brick-work? 

6.  How  is  Grain  usually  estimated?  How  much  does  the  standard 
bushel  contain?  What  is  the  rule  for  finding  the  quantity,  in  bushels, 
of  grain  in  a  bin  or  wagon  ?  For  finding  the  quantity  of  grain  when 
heaped  upon  a  floor? 

7.  How  is  hay  bought  and  sold  ?  About  how  many  cubic  feet  of  clover 
make  a  ton  ?  About  how  many  cubic  feet  of  meadow-hay  make  a  ton  ? 
What  is  the  rule  for  finding  the  quantity  of  hay  in  tons  ? 

8.  What  is  Gauging  ?     What  are  the  rules  for  gauging  ? 


298 


METRIC   SYSTEM. 


SECTION  LXXII. 

METRIC  SYSTEM. 

756c  The  Metric  System  is  a  system  of  weights  and  meas 
ures  based  upon  a  unit  called  a  meter. 

757.  The  Meter  is  one  ten-millionth  part 
of  the  distance  from  the  equator  to  either 
pole,  measured  on  the  earth's  surface  at 
the  level  of  the  sea. 

758.  The  Jfames  of  derived  metric  de- 
nominations are  formed  by  prefixing  to 
the  name  of  the  primary  unit  of  a  meas- 
ure— 

Milli  (mill'e),  a  thousandth;  Centi 
(sent'e),  a  hundredth;  Deci  (des'e),  a 
tenth ; 

Deka  (dek'a),' ten ;  Hecto  (hek'to),  one 
hundred  ;  Kilo  rkil'o),  a  thousand  ;  Myria 
(mir'ea),  ten  thousand. 

This  system,  first  adopted  by  France,  has  been 
extensively  adopted  by  other  countries,  and  is 
much  used  in  the  sciences  and  the  arts.  It  was 
legalized  in  1866  by  Congress  to  be  used  in  the 
United  States,  and  is  already  employed  by  the 
Coast  Survey,  and  to  some  extent  by  the  Mint 
and  the  General  Post-OfEce. 

The  illustration  adjoining  shows  the  length  of 
10  centimeters,  or  a  tenth  of  a  meter,  compared 
with  4  inches,  or  a  third  of  afoot. 

The  nickel  5-cent  pieces  are  each  two  hun- 
dredths of  a  meter  in  diameter;  hence,  50  of 
them  placed  side  by  side  in  a  straight  line  will  measure  1  meter. 

The  simplicity  and  utility  of  the  system,  recognized  now  by  all  civil- 
ized nations,  is  likely  to  lead  to  its  general  adoption,  and  to  the  great 
advantage  of  home  and  foreign  trade. 

In  the  tables  the  units  most  used  are  denoted  by  CAPITALS  or  by  plain 
Roman  type. 


METRIC  SYSTEM.  299 


LINP]AR   MEASURES. 
759.  The  Meter  is  the  primary  unit  of  lengths. 

TABLE. 


10  millimeters  [mm.)      are    1  centimeter 

[cm.) 

= 

.S9S7  in. 

10  centimeters 

'     1  decimeter 

= 

S.937    " 

10  decimeters 

'       1  METER    (m.) 

= 

89.37      " 

10  meters 

'     1  dekameter 

= 

S9S.7 

10  dekameters 

'     1  hectometer 

= 

S28 

ft- 

1            " 

10  hectometers 

'       1  KILOMETER 

{km.) 

= 

.62137  mi 

10  kilometers 

'     1  viyriaincter 

= 

6.2137     " 

The  Meter  is  used  in  ordinary  measurements;  the  Centimeter,  or 
Millimeter,  in  reckoning  very  small  distances ;  and  the  Kilometer,  for 

roads  or  great  distances. 

A  Ceniimeter  is  about  |  of  an  inch ;  a  Meter  is  about  3  feet  3  inches 
and  I  of  an  inch ;  a  Kilometer  is  about  200  rods,  or  f  of  a  mile. 

SURFACE  MEASURES. 

760.  The  Square  Meter  is  the  primary  unit  of  ordinary  sur- 
faces ;  and, 

761.  The  Are  (air),  a  square  each  of  whose  sides  is  ten 
meters,  is  the  unit  of  land  measures. 

TABLE. 

100  sq.  millimeters  (sq.  mm.)  are  1  sq.  centimeter  {sq.  cm.)  =        .155  sq.  in. 

100  sq.  centimeters  "   1  sq.  decimeter  =   15.5  sq.  in. 

f  1550  sq.  in.,  or 
100  sq.  decimeters  "    1  sq.  meter  [sq.  m.)        =  <  ^  ^^^  ^        , 

Also, 

100  centiares,  or  sq.  meters,  are  1  are  (ar.)        =  119.6  sq.  yd. 
100  ares  "    1  hectare  {ha.)  =     2.Jfll  acres. 

A  Square  Meter,  or  1  Centiare,  is  about  10|  square  feet,  or  1^  square 
yards,  and  a  Hectare  is  about  1\  acres. 

CUBIC  MEASURES. 

762.  The  Cubic  Meter,  or  Stere  (stair),  is  the  primary  unit 

of  a  volume. 

TABLE. 

1000  cu.  millimeters  {cu.  mm.)  are  1  cu.  centimeter  {cu.  cm.)  =      .061  cu.  in. 
1000  cu.  centimeters  '•    1  cu.  decimetet  =  61.022  cu.  in. 

1000  cu.  decimeters  "    1  cu.  meter  (cm.  m.)  =  35.314  cu. ft. 


300 


METRIC  SYSTEM. 


The  Stere  (stair)  is  the  name  given  to  the  cubic  meter  in  measuring 
wood  and  timber.  A  tenth  of  a  stere  is  a  Decislere,  and  ten  steres  are  a 
Dekastere. 

A  Cubic  Meter,  or  Stere,  is  about  1  \  cubic  yards,  or  about  2i  cord  feet. 

LIQUID  AND  DRY  MEASURES. 

763.  The  Liter  (leeter)  is  the  primary  unit  of  measures  of 
capacity,  and  is  a  cube,  each  of  whose  edges  is  a  tenth  of  a 
meter  in  length  ;  and, 

764.  The  Hectoliter  is  the  unit  in  measuring  large  quanti- 
ties of  grain,  fruits  roots  and  liquids. 

TABLE. 

10  miUili Icrs  f  m/ . )  a?-e  i  e e  n  t  i  i  i  f  c  r  ( c^. )      =      .888  fl.  oz. 
10  centiliter.^  "    1  deciliter  -=      .845  liq.  gill. 

10  deciliters  "    1  liter  (l.)  =  1.0567  liq.  qt. 

10  liters  "    1  dekaliter  =  2.6417  gal. 

10  dekaliters  "    i  iikctof.itrr  (W.)  =  2bu.  8.35pk. 

10  hectoliters  "    1  kihlitcr  =  28  hu.  ij  pk. 

A  Centiliter  is  about  ^  of  a  fluid  ounce;  a  Liter  is  about  l^j  liquid 
quarts,  or  -f^  of  a  dry  quart ;  a  Hectoliter  is  about  2|  bushels ;  and  a 
Kiloliter  is  1  cubic  meter  or  stere. 


WEIGHTS. 

765.  The  Gram  is  the  primary  unit  of 
weights,  and  is  the  weight  in  a  vacuum  of  a 
cubic  centimeter  of  distilled  water,  at  the  tem- 
perature of  39.2  degrees  Fahrenheit. 


A  CnBic  Ckntimetf.r. 


TABLE. 

10  milligrams  img.)  arc  1  centigra^n 


10  centigrams 
10  decigrains 
10  grams 
10  dekagrams 
10  hectograms 
10  kilograms 
10  myriagrams 
10  quintals 


1  decigram 

"  1  ORAM  (g.) 

"  1  dekagram 

"  1  hectogram 

"  1  KILOGRAM  (k.) 

"  1  myriagram 

"  1  quintal 


.1543  gr.  T. 
UJfS       " 
15.432      " 

.3527  av.  oz. 
3.5274     " 
2.2046  av.  lb. 
22.046 
220.46         " 


2  TO  N  N  K  A  u  ( i. )      =  2204.6 


METRIC  SYSTEM.  301 

The  Grnm  is  used  in  weighinsj  gold,  jewels,  letters  and  small  quantities 
of  things.  Tlie  Kilogram,  or,  for  hrevity.  Kilo,  is  used  by  grocers ;  and 
the  Tonneuit  (tonno),  or  Metric  'Ton,  is  used  in  hnding  the  weight  of  very 
lieavy  articles. 

A  Gram  is  about  15J  grains  Troy;  the  Kilo,  about  2^  pounds  avoirdu- 
pois ;  and  the  Metric  Ton,  about  2205  pounds. 

A  Kilo  is  the  weight  of  a  liter  of  water  at  its  greatest  density,  and  the 
Metric  Ton  of  a  cubic  meter  of  water. 

766.  Metric  Numbers  are  written  with  the  decimal  poiut  (.) 
at  the  right  of  the  figures  denoting  the  unit. 

Thus,  15  meters  3  centimeters  are  written  15.03  m. 

767.  When  metric  numbers  are  expressed  by  figures,  the 
part  of  the  expression  at  the  left  of  the  decimal  point  is  read 
as  the  number  of  the  unit,  and  the  part  at  the  right,  if  any,  as 
a  number  of  the  lowest  denomination  indicated,  or  as  a  decimal 
part  of  the  unit. 

Thus,  46.525  m.  is  read  46  metres  and  525  millimeters,  or  46  and  525 
thousandths  meters. 

768.  In  writing  and  reading  metric  numbers,  according  as 
the  scale  is  10,  100  or  1000,  each  denomination  should  be 
allowed  one,  two  or  three  orders  of  figures. 

WRITTEN  EXERCISES. 

769. — Ex.  1.  Express  by  figures  two  kilometers  one  hundred 
sixty-nine  meters  seventy-five  centimeters  as  meters. 

Ans.  2169.75  m. 

2.  Express  nine  hundred  sixteen  millimeters  as  a  decimal 
of  a  meter.  Ans.  .916  m. 

3.  Express  four  hundred  fifty  kilometers  three  hundred 
twelve  meters  as  kilometers.  Ans.  450.312  km. 

4.  Express  thirty-eight  hectares  three  ares  ninety-four  centi- 
ares  as  hectares.  Ans.  38.0394  ha. 

5.  Express  twenty-five  square  meters  seventy-one  sc^uare 
centimeters  as  square  meters.  Ans.  25.0071  sq.  m. 

6.  Express  five  cubic  meters  one  thousand  seventy-six  cubic 
centimeters  as  cubic  meters.  Ans.  5.001076  cti.  m. 

20 


302 


METRIC  SYSTEM. 


7.  Express  four  hundred  twenty-two  kilos  thirty-five  grams 
as  kilos.  Ans    422.035  k. 

8.  Express  one  hundred  one  tonneaux  nine  hundred  nine 
kilos  as  tonneaux.  Ans.  101.909  t. 

9.  Express  fifty-five  liters  five  centiliters  as  liters. 

Ans.  55.05  i. 

10.  Express  one  thousand  thirty-seven   hectoliters  twenty- 
five  liters  as  hectoliters.  >  Ans.  1037.25  hi. 


Write  and  read — 


11.  2169.75  m. 

12.  195.007  km. 

13.  31.9  cm. 

14.  8.0394  ha. 


15.  104.147  cu.  cm. 

16.  106.07  St. 

17.  51.001001  cu.  m. 

18.  31.15  hi. 


19.  67.3051. 

20.  6.005  gr. 

21.  316.08  k. 

22.  163.455  t. 


COMPUTATIONS. 

770.  The  Computations  in  metric  numbers  are  similar  to 
those  in  United  States  money. 

From  the  nature  of  the  scales,  the  numbers  are  operated 
with  in  like  manner  as  are  simple  integers  and  decimals. 

Thus,  16.55  m.,  or  16  meters  55  centimeters,  may  be  changed  to  centi- 
meters by  removing  the  decimal  point  two  orders  to  the  left,  in  the  same 
manner  as  16.55  dollars  may  be  clianged  to  cents;  and  1634  millimeters 
may  be  changed  to  meters  by  pointing  off  three  orders  from  the  right,  in 
the  same  manner  as  1634  mills  may  be  changed  to  dollars. 

771.  Units  of  the  common  system  may  be  readily  changed 
to  units  of  the  metric  system  l)y  aid  of  the  following 


TABLE. 

1  inch 

=  S.54  centimeters. 

1  cu.  inch 

=  16.89  cu.  centimeters. 

1  foot 

=  S0./f8  centimeters. 

1  cu.  foot 

=  28820  cu.  centimeters 

1  yard 

=  .9144.  meter. 

1  cu.  yard 

=  .7646  cu.  meter. 

1  rod 

^  5.029  meters. 

1  cord 

=  8.635  steres. 

1  mile 

=  1.6098  kilometers. 

1  fl.  ounce 

=  2.958  centiliters. 

1  sq.  inch 

=  6.4528  sq.  centimeters. 

1  gallon 

=  8.786  liters. 

1  sq.  foot 

=  929  sq.  centiJiieters. 

1  bushel 

=  .3524  hectoliter. 

1  sq.  yard 

=  .8861  sq.  meter. 

1  grain  Troy 

=  64.S  milligrams. 

1  sq.  rod 

=  25.29  centiares. 

1  pound  Troy 

=  .873  kilo. 

1  acre 

=  40.47  ares. 

1  pound  av. 

=  .4586  kilo. 

1  sq.  mile 

=  269  hectares. 

1  ton 

=  .907  tonneau. 

METRIC  SYSTEM.  303 

I'RO  BI^EMS. 

1.  Reduce  14937  meters  to  kilometers.         Ans.  14.937  k. 

2.  Reduce  160000  square  meters  to  hectares.     Ans.  16  ha. 

3.  What  decimal  of  a  tonneau  are  83000  grams  ? 

4.  How  many  centiliters  are  56.55  hectoliters  ? 

5.  Reduce  20  miles  40  rods  to  kilometers. 

1.6093  km.  X20^  32.186  km.  SoLUTioN.-Since  l 

.005029"  x4o=  .soiieyn.  ;":,;;  r^'ui: 

32.38716    "  1.6093  km.,  or  32.186 

km.  Since  1  rod  is 
5.029  m.,  or  .00)029  km.,  40  rods  are  40  times  .005029  km.,  or  .20116  km. 
32.186  km.  +  .20110  km.  are  32.38716  km.,  the  result  required. 

6.  How  many  miles  are  32.3871  kilometers  ? 

7.  The  length  of  the  tunnel  through  Mt.  Cenis  is  about 
12.22  kilometers  ;  what  is  its  length  in  miles? 

8.  The  French  post-oflBce  allows  7.5  grams  for  a  single  post- 
age— the  United  States,  \  of  an  ounce  avoirdupois.  How^  many 
grains  Troy  does  the  latter  exceed  the  former?      Ana.  103.01 

9.  How^  many  cubic  centimetres  are  31.631  cubic  meters? 

An%.  31631000. 

10.  How  many  hectares  in  a  rectangular  farm  whose  length 
is  1500  meters  and  width  800  meters ;  and  what  is  its  value  at 
$80  per  acre?  Ans.  120  ha. ;  value,  $23721.60. 

11.  A  square  mile  is  how  many  hectares? 

12.  If  a  .'^ack  of  flour  of  150  kilos  be  sold  at  54.17  francs, 
what  "would  be  the  corresponding  price  of  a  cental  in  United 
States  money,  allowing  5.14  francs  to  a  dollar  ? 

Ans.  $3.18  +  . 

13.  A  bin  is  3.75  meters  long,  2.50  wide  and  1.80  deep. 
How  many  hectoliters  will  it  contain  ? 

14.  What  must  be  the  height  of  a  range  of  wood  nhich  is 
25  meters  long,  1.12  meters  wide,  to  contain  35  steres? 

Ans.  1.25  meters. 

15.  When  wine  is  at  2  francs  a  liter,  what  is  it  a  gallon  in 
United  States  money,  the  value  of  a  fx-anc  being  $.18y^^? 

Ans.  $1.40+. 


304  SERIES  OR   PROGRESSION. 

SECTION    LXXIII. 

SERIES  OP,  PROGRESSION. 

772.  A  Series,  or  Progression,  is  a  succession  of  numbers  in 
which  each  succeeding  number  is  formed  from  the  preceding 
one  by  adding  or  subtracting  the  same  quantity,  or  by  the 
multiplication  by  a  constant  factor. 

The  Terms  of  a  series  are  its  numbers ;  the  first  and  last  terms  are  its 
Extremes,  and  the  other  terms  its  3feans. 

774.  A  series  or  progression  is  ascending  when  the  terms  in- 
crease regularly  from  the  first,  and  descending  when  the  terms 
decrease  regularly  from  the  first. 

ARITHMETICAL  PROGRESSION. 

775.  An  Arithmetical  Prog'ression  is  a  series  whose  terms  in- 
crease or  decrease  by  a  common  difierence. 

Thus,  3,  5,  7,  9,  11,  and  16,  14,  12,  10,  are  arithmetical  progressions  in 
which  the  common  difference  is  2. 

776.  The  first  term,  the  last  term,  the  common  difference, 
the  number  of  the  terms,  and  the  sum  of  the  terms,  are  the 
elements  of  a  series. 

The  relations  of  these  are  such  that  when  three  of  them  are 
known  the  others  may  be  determined. 

WRITTEN  EXERCISISS. 

777. — Ex.  1.  The  first  term  of  an  ascending  arithmetical 
series  is  5,  and  the  common  difierence  2.     What  is  the  4th 

term? 

Solution. 
1st  term  =  5 

Sd      "    =5-¥2  =5 +  2X1  =  1st  term  +  com.  dif.  X  1=    7 

Sd      "     =5  +  2  +  2         =  5  +  2X2=  Ut  term  +  com.  diff.  X2=    9 

4/h     "     =5  +  2  +  2  +  2  =  5  +  2X3=  1st  term  t  com.  dif.  \S=  11 

2.  The  first  term  of  a  descending  arithmetical  series  is  11, 
and  the  common  difi"orGnce  2.     Wliat  is  the  4th  term  ? 


SEEIES  OB  PJiOGIii:SSION.  305 

Solution. 
1st  term  =  11 

U       "     -^  11-3  =11  —  {2X1)  =  1st  term  —  com.  diff.  X  1 -^  9 

Sd       '-.    =11  —  2  —  2        =11  —  {2X  2)  =  1st  ter}ii  —  com.  diff.  X  2  -=  7 

4lh     "     =  11  — 2  — 2  — 2=  11  — [2X8)  =  1st  term  —  com.  dif.  ,<  ,i  =  5 

3.  What  is  the  sum  of  the  arithmetical  series  3,  7,  11,  15, 

19? 

Solution. 

3,      7,     11.     lo,     19,  w  the  arithmetical  series. 

19,     15,     11,      7,      3,  is  the  series  inverted. 

22  +  22  +  22  +  22  ^  22  =  110,  the  sum  of  twice  the  scries. 

11  +  11  +  11  +  11  +  11  —    55,  the  sum  of  the  series. 

Or,  since  the  sum  of  the  extremes,  or  of  any  two  terms  equally  distant 
from  them,  is  the  average  of  the  several  term.s  of  the  series,  ^  .^  ^  X  5 
=  55,  is  tlie  sum  of  the  series. 

778.  Rules  for  Arithmetical  Progression.— i.  Multiply  the 
coimnon  dif}x'rencc  by  the  niunber  of  terms  less  one; 
add  the  product  to  the  smaller  extreme,  and  the  sum 
will  he  the  greater ;  or,  subtract  the  product  from  the 
greater  extreme,  and  the  remainder  will  be  the 
smaller. 

2.  Multiply  half  the  sum  of  the  extremes  by  the 
number  of  terms,  and  the  result  will  be  the  surn  of 
the  series. 

rROBLEMS. 

1.  A  man  being  asked  the  age  of  his  eldest  child,  replied 
that  his  youngest  child  was  2  years  old,  the  number  of  his 
children  was  6,  and  the  common  difference  in  their  ages  was 
3  years.     What  was  the  age  of  the  eldest  child  ? 

Ans.  17  years. 

2.  A  man  travelled  10  days,  increasing  the  distance  gone 
over  H  niiles  each  day  compared  with  the  day  preceding,  so 
that  he  travelled  the  tenth  day  24  miles.  How  far  did  he 
travel  the  first  day?  Ajis.  10^  miles. 

3.  What  will  $400,  at  7%  simple  interest,  amount  to  in  44 
years?  A^is.  $1632. 

26* 


306  SEBIES   OR   PBOGJRESSIOir. 

4.  There  are  a  number  of  rows  of  corn,  the  first  of  which 
contains  3  hills,  the  second  7,  the  third  11,  and  so  on  to  the 
last,  which  has  43  hills.  How  many  hills  are  there  in  all  the 
rows?  Ans.  253. 

5.  If  a  stone  fall  through  16.1  feet  in  the  first  second,  48.3 
feet  in  the  second  second,  80.5  feet  in  the  third  second,  and  so 
on,  how  deep  will  be  the  shaft  of  a  mine  where  a  stone  takes  7 
seconds  to  reach  the  bottom  ? 

6.  If  you  should  begin  witli  a  capital  of  $3500,  and  decrease 
every  year  $60,  what  would  remain  of  your  capital  at  the  end 
of  10  years  ? 

GEOMETRICAL  PROGRESSION. 

"JIO.  A  Geometrical  Progression  is  a  series  whose  terms  in- 
crease or  decrease  by  a  constant  factor. 

780.  The  Rate,  or  Ratio,  of  a  series  is  the  constant  factor. 

Thus,  3,  6,  12,  24,  48,  is  a  geometrical  series  whose  rate  is  2 ;  and  75, 
15,  3,  is  a  geometrical  series  whose  rate  is  ^. 

781.  An  Infinite  Scries  is  a  descending  series  of  an  infinite 
number  of  terms. 

Of  such  a  series  the  last  term  must  be  smaller  than  any 
assignable  quantity ;  hence,  it  may  be  considered  0. 

782.  As  in  an  arithmetical  series,  the  relation  of  the  ele- 
ments of  a  geometrical  series  are  such  that  when  any  three  of 
them  are  known  the  others  may  be  determined. 

WRITTEN  EXEJtCISES. 

783. — Ex.  1.  The  first  term  of  a  geometrical  series  is  5,  and 
the  common  rate  2.     What  is  the  fourth  term  of  the  series  ? 

SOLUTIOX. 

1st  term  =  g 

£d     "      =5X2  =5X2^  =-  1st  term  X  rate  =  10 

Sd     "      =5X2X2  =  5X2^==  1st  term  X  square  of  the  rate  =--  20 

4th    '•      =5X2X2X2  =  5X2^  =  1st  term  X  cube  of  the  rate     =  40 

2.  What  is  the  sum  of  a  geometrical  series  whose  first  term 
is  5,  last  term  135,  and  rate  3? 


SEIiIi:S   OB   FEOGBESSIO^^.  307 

Solution. 

15  +  45  +  135  +  405  =  3  times  the  tium  of  the  series. 
5  +  15  +  4^  +  135  =  once  the  .siun  0/  the  series. 

—  5  +  405  =  twice  the  sum  of  the  scries. 

Hence,  ^,r^  ~  ^^'^  "^  ^^^'^  *""*■  °/  '^''^  series. 

Had  the  series  been  descending,  the  first  term  135,  and  rate  \,  by 
inverting  the  series,  making  the  first  term  the  last,  and  the  fate  3,  tlie 
sokition  would  then  be  the  same  as  now  given. 

3.  What  is  the  sum  of  au  infinite  series  whose  first  term  is  4, 

and  rate  \  ? 

Solution. 

The  series  extended  =-4,  1,  r'  t,'  •  ■  •  0. 
U      16 

The  series  inverted  =  0  .  .  .  —>  —>  1,  4> 


16    U 


whose  rate  is  4.     Then, 


4  times  the  sum  of  the  series  ^  ^  ...+-+  -  +i  +  ^  +  -Z5 
Once  the  sum  of  the  scries      ^  ^  •  •  '  ~^  Jg~^  7  +1  +  4 


3  times  the  sum  of  the  series  =  16 

Hence,  the  sum  of  the  series  =  --  =  5— . 

784.  Rules  for  Geometrical  Progression.— i.  Multiply  the 
first  tervh  by  the  rate  raised  to  a  power  whose  expo- 
nent is  one  less  than  the  nmnher  of  terms,  and  the 
product  ivill  he  the  required  term. 

2.  Multiply  the  last  term  hy  the  rate;  subtract  the 
first  term  from,  the  product,  and  divide  the  differ- 
ence by  the  rate  less  one,  and  the  result  ivill  be  the 
sum  of  the  series. 

If  the  series  is  descending,  use  the  series  inverted, 
maldng  the  first  term  the  last,  and  the  rate  greater 
than  one;  and, 

If  the  series  is  infinite,  multiply  the  larger  term,  by 
the  rate  of  the  series  inverted,  and  divide  the  quotient 
by  the  rate  less  one. 


308  LIFE   INTERESTS   AND   REVERSIONS. 

PROJiT^EMS. 

1.  A  person,  travelling,  goes  5  miles  the  first  da)',  10  miles 
the  second,  20  miles  the  third,  and  so  on.  If  he  travel  7  days, 
how  far  will  he  go  the  last  day  ?  Ans.  320  miles. 

2.  What  is  the  amount  of  6100  for  9  years,  at  0%  compound 
interest?  Ans.  $155.13. 

Here  Si 00  is  the  first  term,  10,  or  1  more  than  the  number  of  years, 
is  the  number  of  terms,  and  1.05  is  the  rate.     Rr^'inired  the  last  term. 

8.  What  is  the  sum  of  the  series  2,  1,  \,  \,  etc.,  to  infinity? 

Am.  4. 

4.  If  a  ball  be  put  in  motion  by  a  force  which  would  move 
it  10  rods  the  first  minute,  8  rods  the  second,  6.4  the  third, 
and  so  on  in  the  ratio  of  .8,  how  far  would  it  move? 

5.  If  a  farmer  should  sow  5  grains  of  wheat,  and  its  produce 
every  year  for  9  years,  how  many  bushels  would  there  be  in 
the  last  harvest,  supposing  that  each  harvest  amounts  to  10 
times  the  quantity  sov/ed,  and  that  8000  grains  make  1  pint  ? 

Ans.  9765  bu.  2^  pk. 


SECTION  LXXIV. 
LIFE  IJ^TE RESTS  AJs^B  BEYERSIOKS. 

785.  An  Annuity  is  a  sum  of  money  to  be  paid  annually,  or 
at  regular  intervals  of  time. 

786.  A  Life  Interest  is  an  annuity  to  continue  for  life  or 
lives. 

Thus,  a  pension  for  life  and  a  widow's  dower,  or  life  estate,  are  each  a 
life  interest. 

787.  A  RcTcrsioiiary  Intei'cst  is  an  interest  which  does  not 
commence  until  alUr  a  certain  period,  or  until  after  a  certain 
event. 

788.  The  United  States  Treasury  Dci)artment,  in  the  com- 
putation of  life  interests,  uses  the  following  tables,  kno\vu  ass 
the  Carlisle  Tables : 


LIFE  INTERESTS  AND   REVERSIONS. 


309 


CARLISLE  TABLES, 

Of  the  Expectancy  of  Life,  and  of  the  Present    Value  of  Life  Annuities. 


M^. 

B5 

Present  Value  of  An- 
nuity  of  $1   for   the 
years  in  2d  column, 
interest  at  6  per  cent. 

Present  Value  of  $1  to 
be  received  at  the  end 
of  the  years  in  2d  col- 
umn,   interest    at  6 
per  cent. 

Age. 

c  ^ 

1 

B. 
X 

Present  Value  of  An- 
nuity of  $1  for  the 
years  in  2d  column, 
interest  at  6  per  cent. 

Present  Value  of  $1  to 
bo  received  at  the  end 
of  the  years  in  2d  col- 
umn,   interest    at   6 
per  cent. 

0 

38.72 

14.9202 

.104788 

41 

26.97 

13.2043 

.207741 

1 

44.68 

15.4325 

.074065 

42 

26.34 

13.0737 

.215580 

2 

47.55 

15.6225 

.062645 

43 

25.71 

12.9395 

.223635 

3 

49.82 

15.7521 

.054874 

44 

25.09 

12.8032 

.231812 

4 

50.76 

15.8008 

.051953 

45 

24.46 

12.6576 

.240548 

5 

51.25 

15.8252 

.050490 

46 

2.3.82 

12.5059 

.249646 

6 

51.17 

15.8213 

.050722 

47 

23.17 

12.3454 

.259278 

7 

50.80 

'  15.8029 

.051830 

48 

22.51 

12.1751 

.269494 

8 

50.24 

15.7742 

.053551 

49 

21.81 

11.9889 

.280669 

9 

49.57 

15.7385 

.055689 

50 

21.11 

11.7946 

.292324 

10 

48.82 

15.6972 

.058168 

51 

20.39 

11.5846 

.304922 

11 

48.04 

15.6523 

.060860 

52 

19.68 

11.3701 

.317792 

12 

47.27 

15.6055 

.063670 

53 

18.97 

11.1481 

.331108 

13 

46.51 

15.5573 

.066559 

54 

18.28 

10.9201 

.344791 

14 

45.75 

15.5072 

.069566 

55 

17.58 

10.6805 

.359172 

15 

45.00 

15.4558 

.072650 

56 

16.89 

10.4364 

.373815 

16 

44.27 

15.4028 

.075832 

57 

16.21 

10.1839 

.388767 

17 

43.57 

15.3501 

.078996 

58 

15.55 

9.9287 

.404275 

18 

42.87 

15.2956 

.082267 

59 

14.92 

9.6788 

.419268 

19 

42.17 

15.2384 

.085695 

60 

14.34 

9.4368 

.433789 

20 

41.46 

15.1778 

.089331 

61 

13.82 

9.2154 

.447078 

21 

40.75 

15.1151 

.093095 

62 

13.. 31 

8.9900 

.460612 

22 

40.04 

15.0500 

.097002 

63 

12.81 

8.7636 

.474183 

23 

39.31 

14.9972 

.101248 

64 

12.30 

8.5245 

.488530 

24 

38.59 

14.9068 

.105592 

65 

11.79 

8.2793 

.503231 

25 

37.86 

14.8307 

.110157 

66 

11.27 

8.0211 

.518737 

26 

37.14 

14.7521 

.114876 

67 

10.75 

7.7552 

.534690 

27 

36.41 

14.6685 

.119892 

68 

10.23 

7.4813 

.551125 

28 

35.69 

14.5829 

.125024 

69 

9.70 

7.1926 

.568446 

29 

35.00 

14.4982 

.130105 

70 

9.18 

6.9022 

.585867 

30 

34.34 

14.4123 

.135258 

71 

8.65 

6.5945 

.604328 

31 

33.68 

14.3240 

.140560 

72 

8.16 

6.3045 

.621730 

32 

33.03 

14.2343 

.145938 

73 

7.72 

6.0341 

.637953 

33 

32.36 

14.1366 

.151789 

74 

7.. 33 

5.7894 

.652634 

34 

31.68 

14.0344 

.157932 

75 

7.01 

5.5887 

.664681 

35 

31.00 

13.9291 

.164255 

76 

6.69 

5.3762 

.677427 

36 

30.32 

13.8174 

.170957 

77 

6.40 

5.18.33 

.6S8999 

37 

29.64 

13.7021 

.180796 

78 

6.12 

4.9971 

.700193 

38 

28.96 

13.5833 

.185000 

79 

5.80 

4.7763 

.713420 

39 

28.28 

13.4579 

.192530 

80 

5.51 

4.5719 

.725687 

40 

27.61 

i 

13.3289 

.200208 

81 

5.21 

4.3604 

.738376 

SIO  LIFE  INTERESTS  AND  BEVERSIONS. 

PROHrUMS. 

1.  What  is  the  present  value  of  a  "widow's  dower  whose 
yearly  rent  is  $520,  and  whose  age  is  49  years  ? 

Solution. 
Expectancy  of  life  at  Jf9  years  of  age  =  21. 81  years. 

Present  value  of  >$1  annuity  for  21.81  years  =  $11.9889. 
Present  value  of  $520  annuity  for  21.81 

years  =  $11.9889  X  520  =  $6234-23. 

2.  What  is  the  ready-money  value  of  a  legacy  of  $1000,  to 
be  received  after  35  years  ? 

Solution. 
Present  value  of  $1,  to  be  received  after  35  years  =  $.130105. 
Present  value  of  $1000,  to  be  received  after  35 

years  =  $.130105  X  1000  =  $130.10\ 

3.  A  person  62  years  old  has  a  yearly  pension  of  896  ;  what 
is  its  present  value  ?  Ana  $863.04. 

4.  A  widow  aged  51  has  set  off,  as  her  dower,  property  whose 
appraised  value  is  $3800.  What  is  the  present  value  of  the 
reversionary  interest  ?  Ans.  $1158.70. 

5.  Smith,  who  is  70  years  old,  has  a  life  annuity  of  S700  per 
annum.     What  is  its  present  value? 

6.  AMiat  should  be  the  present  value  of  a  legacy  of  $4000, 
to  be  received  after  10  years  9  months  ? 


TEST    QUESTIONS. 

789. — 1.  What  is  the  Metric  System?  A  meter?  How  are  the 
names  of  derived  metric  denominations  formed  ? 

2.  What  is  the  Primary  Unit  of  lengths?  Of  ordinary  surfaces? 
Of  land  measures?  Of  volumes?  Of  capacity  ?  Of  large  quantities 
of  grains,  fruits,  etc.? 

3.  How  are  Metric  Numbers  written  ?  How  many  orders  of  figures 
are  allowed  to  each  metric  denomination  ?  How  is  the  part  of  a  metric 
expression  at  the  left  of  the  decimal  point  read?  At  the  right  of  the 
decimal  point? 


PROBLEMS    FOR    ANALYSIS.  311 

4.  What  is  a  Skries  or  a  Progression  ?  What  are  the  terms  of  a 
series?  The  extremes?  The  means?  When  is  a  series  ascending? 
When  descending? 

5.  What  is  an  Akitiimetical  Progression?  What  is  the  rule  for 
finding  either  extreme  of  an  arithmetical  series  ?  For  iinding  the  sum 
of  the  series  ? 

6.  What  is  a  Geometrical,  Progression  ?  The  rate  or  ratio  of  a 
geometrical  series?  What  is  an  inlinite  series?  What  is  the  rule  for 
finding  the  last  term  of  a  geometrical  scries  ?  For  finding  the  sum  of  an 
ascending  series  ?  The  sum  of  a  descending  series  ?  The  sum  of  an 
infinite  series  ? 

7.  What  is  an  Annuity  ?  A  life  interest  ?  A  reversionary  interest  ? 
What  is  used  by  the  United  States  Treasury  Department  in  the  computa- 
tion of  life  interests? 

8.  How  do  the  metric  measures  differ  from  the  measures  in  common 
use?  How  do  arithmetical  and  geometrical  jirogressions  differ?  How 
does  a  reversionary  interest  differ  from  a  life  interest  ?  How  does  a  root 
differ  from  a  power  ?  The  square  root  from  the  cube  root  ?  Is  the  root 
of  a  proper  fraction  smaller  or  larger  than  its  corresponding  power? 


SECTION  LXXV. 
PROBLEMS  FOR  AJ^ALYSIS. 

MENTAI.   EXERCISES. 

790. — 1.  Two  boys  on  counting  their  marbles  found  that 
one  had  9  more  than  the  other,  and  that  together  they  had 
49.     How  many  had  each  of  the  boys  ? 

2.  The  greater  of  two  numbers  is  7f ,  and  their  difference  is 
3|.     What  are  the  numbers  ? 

3.  A  man  being  asked  how  many  cows  he  had,  answered 
that  if  he  had  2  more,  twice  the  number  woukl  be  26.  How 
many  cows  had  he  ? 

4.  If  John  were  3  years  younger,  twice  his  age  would  be  18 
years.     How  old  is  he  ? 

5.  A  man  wishing  to  contribute  money  to  an  equal  number 
of  poor  men  and  women,  gave  to  each  man  9  dimes  and  to 
each  woman  3  dimes.  If  he  gave  them  in  all  $120,  how  many 
men  and  women  were  there  respectively? 


312  FEOBLEMS  FOR  ANALYSIS. 

6.  If  9  be  taken  from  the  sum  of  two  numbers  of  v/hich  7 
is  one,  there  Aviil  be  13  left.     What  is  the  larger  number? 

7.  A  and  B  by  -working  together  can  do  a  piece  of  work  in 
2  days.  B  can  do  it  alone  in  5  days.  How  much  of  it  can 
both  together  do  in  1  day?  What  part  of  it  can  A  alone  do 
in  1  day  ?    In  what  time  can  A  alone  do  it  ? 

8.  Edward  and  Philip  stai't  from  the  same  place  and  travel 
the  same  road.  Edward  starts  4  days  before  Philip,  and  ti'avels 
20  miles  a  day.  Philip  follows,  travelling  25  miles  a  day.  In 
what  time  vvill  Philip  overtake  Edward  ? 

9.  A  thief,  having  50  steps  the  start  of  an  officer,  takes  4 
«teps  while  the  officer  takes  3 ;  and  2  steps  of  the  officer  are 
equal  to  3  of  the  thief.  How  many  steps  can  the  thief  take 
before  the  officer  can  catch  him  ? 

Solution. — 2  steps  of  the  officer  are  equal  to  3  of  the  thief;  hence, 
6  steps  of  tlie  officer  are  equal  to  9  steps  of  the  thief 

While  the  officer  takes  6  steps,  the  thief  takes  8 ;  hence,  while  the 
thief  takes  8  steps,  the  officer  gains  upon  him  1  step  of  the  thief 

Then  while  the  officer  is  gaining  50  of  the  steps  of  the  thief,  the  thief 
can  take  8  times  50  steps,  or  400  steps. 

10.  An  officer  is  in  pursuit  of  a  thief  Avho  has  some  miles 
the  start.  The  thief  goes  20  miles  a  day,  and  the  officer  25. 
If  it  take  the  officer  8  days  to  overtake  the  thief,  how  many 
days  had  the  latter  the  start  ? 

11.  If  4  men  can  earn  $16  in  2  days,  how  long  will  it  take  6 
men  to  earn  $48  ? 

12.  If  6  men  can  do  a  piece  of  work  in  8  days,  what  num- 
ber of  men  can  do  ^  of  it  in  f  of  the  time  ? 

13.  Arthur  is  IG  years  old,  and  Albert  is  4.  In  how  many 
years  will  Arthur  be  only  twice  as  old  as  Albert  ? 

Solution. —  4  years  ago  Arthur  was  12  years  old;  hence  in  12  years 
from  that  time,  or  in  12  years  less  4  from  this  time — that  is,  in  8  years — 
Arthur  will  be  2-1  years  old,  or  twice  as  old  as  Albert. 

14.  Mary  is  15  years  old,  and  her  mother  36.  In  what  time 
will  Mary  be  only  \  as  old  as  her  mother  ? 

15.  A  father  is  35  years  old,  and  his  son  is  5.  In  what  time 
will  the  son  be  \  as  old  as  his  father? 


PROBLEMS  FOR  ANALYSIS.  313 

16.  John  gave  each  of  his  brothers  4  apples,  giving  all  he 
had.  If  he  had  had  12  more  apples  he  could  have  given  each 
of  his  brothers  7  apples.     How  many  brothers  had  he? 

Solution. — To  give  7  apples  to  each  brother  he  would  have  required 
3  apples  more  for  each;  but  he  would  have  required  12  more  for  all. 
Hence,  there  were  as  many  brothers  as  3  apples  are  contained  times  in 
12  apples,  which  are  4. 

17.  Jane  wishes  to  purchase  a  certain  number  of  oranges. 
If  she  pays  6  cents  each,  she  will  have  10  cents  left ;  but  if 
she  pays  8  cents  each,  it  will  take  all  her  money.  How  many 
oranges  does  she  want  ? 

18.  James  wishes  to  divide  some  peaches  among  his  friends. 
If  he  gives  each  of  them  3  ho  will  have  9  left ;  but  if  he  tries 
to  give  each  of  them  5,  he  will  not  have  enough  by  5.  How 
many  friends  has  he  ? 

19.  Eggs  are  sold  at  the  rate  of  4  for  5  cents.  At  what  rate 
were  they  bought  if  the  profit  is  25  per  cent.  ? 

20.  A  has  5  times  as  much  money  as  B  has,  and  the  sum. 
of  the  interest  received  by  both  for  2  years,  at  7  per  cent.,  is 
$70.     What  sum  has  each  ?    ' 

21.  If  a  wagon  cost  $80,  wdiat  would  be  the  cost  of  a  har- 
ness if  -^  of  the  cost  of  the  wagon  were  ^  of  the  cost  of  the 
harness  ? 

22.  A  has  4  times  as  much  money  as  B  has,  and  the  sum 
of  the  interest  received  by  both  for  2  years,  at  7  per  cent.,  is 
870.     What  sum  has  each? 

23.  A  and  B  have  $12,  and  i  of  A's  money  equals  \  of  B's. 
How  many  dollars  has  each  ? 

Solution. — \  of  A's  money  equals  \  of  B's ;  hence,  f ,  or  the  whole, 
of  A's  money  must  equal  f  of  B's. 

If  A's  money  equals  f  of  B's,  and  B's  must  equal  f  of  itself,  $12  ran^t 
equal  |  plus  f ,  or  f ,  of  B's. 

If  f  of  B's  money  is  $12,  ^  is  i  of  $12,  or  $2,  and  |,  or  A's  money,  is 
$4,  and  |,  or  B's  money,  is  $8. 

24.  A  pole  60  feet  long  broke  into  two  parts,  one  of  which 
was  f  of  the  other.     What  was  the  length  of  each  part  ? 

25.  Edward  savs  to  Thomas,  "  ^-  of  my  age  lacks  2  years 

27 


314  PROBLEMS  FOR  ANALYSIS. 

of  being  |  of  yours,  and  the  sum  of  our  ages  is  33  years." 
What  is  the  age  of  each  ? 

26.  The  time  between  three  and  four  o'clock  is  such  that  f 
of  the  minutes  past  three  are  equal  to  f  of  the  minutes  before 
four.     What  is  the  time  ? 

27.  What  is  the  time  in  the  afternoon  when  the  time  past 
noon  is  equal  to  \  of  the  time  to  midnight  ? 

WRITTEN   EXERCISES. 

791. — Ex.  1.  Two  travellers  leave  the  same  place  at  the 
same  time.  One  goes  20  miles  a  day  and  the  other  23|-.  How 
far  apart  will  they  be  at  the  end  of  28  days,  if  they  both 
travel  in  the  same  direction  ?  How  far  if  they  travel  in  oppo- 
site directions?  Ans.  98  miles;  1218  miles. 

2.  A  merchant,  who  commences  business  with  a  capital  of 
$12000,  gains  at  the  rate  of  69000  in  4  years,  by  trading  in 
flour,  and  at  the  rate  of  S9000  in  6  years,  by  trading  in  grain. 
If  his  annual  expenses  are  $4500,  in  what  time  will  he  have 
lost  all?  ^ns.  16  years. 

3.  I  have  a  stick  of  squared  timber  20  feet  6  inches  long,  16 
inches  wide  and  12  inches  thick.  If  3|-  solid  feet  should  be 
sawed  off  at  one  end,  how  long  would  the  stick  then  be  ? 

7790 ,--,  V?-      ^^,o-  *  Solution. — 

-^■^  =  -If  =  ^^-^  *■"■•  =  ^fi-  H  ^'-        1' 23  solid  incites 

are  1  solid  foot ; 

20  jt.  6  in.  —  2Jt.  7^  in.  =  17  ft  10^  in.         hence,  the  piece 

sawed  off  must 
contain  3J-  times  1728  solid  inches,  or  6048  solid  inches.  The  6048  solid 
inches  must  be  the  ])roduct  of  the  numbers,  in  inches,  which  denote  the 
dimensions  of  the  piece  sawed  off. 

The  product  of  the  width  bv  the  thickness,  or  16  X  12,  is  192;  hence, 
the  quotient  of  6048  divided  by  192,  which  is  31.5,  must  denote  the 
number  of  inches  of  the  ])iece  sawed  off. 

The  stick  with  31.5  inches,  or  2  ft.  7^  in.,  sawed  off  must  be  20  ft.  6  in. 
—  2  ft.  11  in.,  or  17  ft.  101  in.  long. 

4.  From  a  plank  which  is  16  feet  5  inches  long  I  wish  to 
cut  off  a  strip  containing  a  square  yard.  At  what  distance 
from  the  edge  must  the  line  be  drawn  ?      Aiis.  6j-^  inches. 


PROBLEMS  FOR  ANALYSIS.  315 

5.  My  bushel  and  half-peck  measures,  which  are  of  a  cylin- 
drical form,  are  respectively  18|^  and  'd\  inches  in  diameter. 
What  must  be  the  depth  of  each  ? 

6.  I  have  a  range,  56  feet  long,  of  firewood,  cut  4  feet  long. 
When  such  wood  is  worth  $6  per  cord,  how  high  must  the 
range  be  piled  to  be  worth  $52.50  ?  Ans.  5  feet. 

7.  A  laborer  agreed  to  work  12  weeks  upon  the  conditions 
that  he  should  receive  $18  per  week  for  every  week  he  worked, 
and  for  every  week  he  was  idle  he  should  pay  $3.50  for  his 
board.  At  the  expiration  of  the  time  he  received  $151.50. 
How  many  weeks  did  he  work  ? 

$  18 y.  12  =^$'2 16  Solution.  —  Had  he  labored 

$2i6-$i5i.5o=$64.5o  ^ z^^:,:T,^.::t:t 

$18  +  $3.50  =  $21.50       ceived  but  $151.50,  he  lost  by  idle- 
$64.50-^$21.50  =  3  ness  $216- $1 51.50    or  $64.50. 

^  Each  week  he  was  idle  he  lost  his 

12  weeks— J  weeks  =  9  weeks.       ""^^ges  and   $3.50,  amounting  to 

$21.50;  hence  he  was  idle  as 
many  weeks  as  $21.50  is  contained  times  in  $64.50,  or  3  weeks.  Since 
he  was  idle  3  weeks,  he  worked  12  weeks  less  3  weeks,  or  9  weeks. 

8.  James  received  $1  a  day  for  his  work,  and  paid  $.25  for 
every  day  he  was  idle.  At  the  end  of  18  days  he  received 
$6.75.     How  many  days  was  he  idle  ? 

9.  A  man  agreed  to  carry  28  packages  to  a  certain  place  on 
the  conditions  that  for  every  one  promptly  delivered  he  should 
receive  30  cents,  and  for  every  one  delayed  he  should  forfeit  50 
cents.  He  forfeited  $2.80  more  than  he  received.  How  many 
packages  did  he  deliver  promptly?  Ans.  14. 

10.  How  many  cows  can  be  kept  on  a  farm  of  48  acres  if 
for  every  5  cows  there  must  be  2  acres  of  meadow,  and  for 
every  3  cows  2  acres  of  pasture-land  ? 

2       2        16  SoiiUTlON. — Since  for  every  5  cows  there  must  be 

5    '   J       25^        2  acres  of  meadow,  and  for  every  3  cows  there  must 

j„  be  2  acres  of  pasture,  for  1  cow  there  must  be  f  of  an 

.48  ~^  jY  =  -40       acre  of  meadow  and  f  of  an  acre  of  pasture,  or  j-f 

of  an  acre.     Hence,  as  many  cows  can  be  kept  on  a 

farm  of  48  acres  as  48  contains  times  xf ,  or  45. 


816  PROBLEMS  FOR  ANALYSIS. 

11.  A  farmer  has  150  acres.   He  cultivates  5  acres  for  every 

3  horses  he  has,  and  allows  10  acres  of  pasture  for  every  4 
horses.     How  many  horses  can  he  keep  ?  Ans.  36. 

12.  A  certain  field  will  furnish  pasturage  for  3  horses,  or  4 
cows,  56  days.  For  what  time  will  it  furnish  pasturage  for  1 
horse  and  1  cow  grazing  together?  Ans.  96  days. 

13.  A  bankrupt's  stock  was  sold  for  $1660,  at  a  loss  of  17% 
on  the  cost  price.  Had  it  been  sold  in  the  course  of  trade,  it 
would  have  realized  a  profit  of  20%.  How  much  was  it  sold 
below  the  trade  price  ?  Ans.  ^740. 

14.  A  merchant  sells  tea  to  a  trader  at  a  profit  of  60%  ;  but 
the  trader  becomes  bankrupt  and  only  pays  75  cents  on  a 
dollar.     How  much  per  cent,  does  the  merchant  gain  or  lose  ? 

Ans.  He  gains  20%. 

15.  The  head  of  a  fish  is  28  inches  long ;  the  tail  is  as  long 
as  the  head  and  |  of  the  body ;  and  the  body  is  as  long  as  the 
head  and  tail.     What  is  the  length  of  the  fish  ? 

28  in.  +  28  in.  =    56  in.  =  j  the  length       ,   Solution.  -  Since 

^        f  1     I         the  tail  IS  as  long  as 

''  ^'      tlie  head  and  ^  of  the 

S6  in.  X  .^  =  22.^  in  =  the  length  of     body,  and  the  body  is 

the  fish.  as   long   as    both   the 

head  and  tail,  28  inches 

plus  28  inches,  or  56  inches,  must  equal  \  the  length  of  the  body.    Since 

the  body  is  \  the  length  of  the  fish,  56  inches  must  equal  \  the  length  of 

the  fish,  and  56  inches  X  4,  or  224  inches,  must  equal  the  length  of  the 

fish. 

16.  A  sum  of  money  was  divided  among  3  men.  The  first 
received  $96,  the  second  received  half  as  much  as  the  third, 
and  the  third  received  as  much  as  the  other  two.  How  much 
did  the  second  and  the  third  receive  ? 

17.  A,  B  and  C,  dividing  a  plantation  consisting  of  120 
acres,  agreed  that  B  should  have  a  third  part  more  than  A, 
and  C  a  fourth  part  more  than  B.  What  number  of  acres 
will  each  have?  Ans.  A,  30  ;  B,  40 ;  C,  50. 

18.  If  a  steamboat,  running  uniformly  at  the  rate  of  12  miles 
per  hour  in  still  water,  were  to  run  4  hours  with  a  current  of 

4  miles  per  hour,  then  to  return  against  that  curi-ent,  what 


PROBLEMS  FOR  ANALYSIS.  317 

length  of  time  from  the  time  she  started  would  she  require  to 
reach  the  place  whence  she  started  ?  Ans.  12  hours. 

19.  A  merchant  purchased  goods  for  $1200,  and  sold  them 
at  a  loss  of  12^%.  He  then  purchased  more  goods  with  the 
proceeds,  and  sold  them  at  a  gain  of  14%.  Did  he  gain  or 
lose  by  these  transactions,  and  how  much  ? 

Ans.  Lost  $3, 

20.  I  wish  to  plant  5292  trees  equally  distant  in  straight 
rows,  and  to  make  the  length  of  the  grove  3  times  the  width. 
How  many  of  the  shorter  rows  shall  I  have  ? 

Solution.  —  Since  the  length  is  3  times  the 
width,  one  third  of  the  trees  are  to  form  an  exact 
square.  The  square  root  of  one  third  of  the 
number  5292  is  42,  which  must  denote  the  number 

of  trees  in  a  side  of  the  square.     Since  there  must  be  3  such  squares, 

there  must  be  3  times  42,  or  126,  short  rows. 

21.  A  farmer  has  2  10-acre  lots ;  one  is  a  square,  and  the 
other  is  a  parallelogram  4  times  as  long  as  it  is  wide.  How 
many  rods  of  fence  will  each  lot  require  to  exactly  enclose  it  ? 

Ans.  The  square,  160  rods ;  the  parallelogram,  200 
rods. 

22.  What  is  the  mean  proportional  between  4  and  9  ? 

Ay^  9  =  36  Solution. — Since  the  mean  proportional  between 

/  q/^  __  /J  the   extremes   of  a   proportion  is  one   of  the   equal 

means  of  the  proportion  (Art.  547),  the  mean  pro- 
portional between  two  numbers  is  equal  to  the  square  root  of  the  product 
of  those  numbers.  The  product  of  4  by  9  is  36,  and  the  square  root  of 
36  is  6,  the  mean  proportional  required. 

23.  What  is  the  mean  proportional  between  7  and  252  ? 

Ans.  42. 

24.  A  cheese,  when  put  into  one  scale  of  an  incorrect  balance, 
was  found  to  weigh  31^  pounds,  but  when  put  into  the  other  it 
weighed  only  20  pounds.     What  was  its  true  weight  ? 

Ans.  25  pounds. 

25.  I  have  corn  of  four  different  qualities,  worth  respectively 
65,  72,  80  and  90  cents  per  bushel.     In  what  proportions  may 

27* 


318  PROBLEMS  FOR   ANALYSIS. 

these  kinds  be  taken  to  form  a  quantity  worth  75  cents  per 

busliel  ? 

Solution. 

At  65  c.  to  gain  1  c.  take  -—  bu. ;  --  hu,  X  10^=  Ibu. 
"72c.  "  Ic.  "  ^  hu.;  j  bu.XlS  =  5bu. 
"  80c.  to  lose  Ic.    "     -  hu.;    ^  hu.XlO  =  2bu. 

5  5 

"90c.      "       Ic.    "    ~hu.;   4^bu.X15=lbu. 

15  15 

On  1  bushel,  worth  65  cents,  taken  at  75  cents,  there  is  a  gain  of  10 
cents ;  hence,  to  gain  1  cent  we  take  y^j  of  a  bushel.  On  1  bushel  at  80 
cents  there  is  a  loss  of  5  cents ;  hence,  to  lose  1  cent,  we  take  \  of  a 
bushel.  Therefore,  we  take  j'^  of  a  bushel  at  65  cents  as  often  as  we 
take  5  of  a  bushel  at  80  cents ;  or,  multiplying  these  fractions  by  the 
least  common  multiple  of  their  denominators,  we  find  we  may,  also, 
take  1  bushel  at  65  cents  as  often  as  we  take  2  bushels  at  80  cents. 

In  like  manner  we  take  \  of  a  bushel  at  72  cents  as  often  as  we  take  -^ 
of  a  bushel  at  90  cents ;  or  we  may  take  5  bushels  at  72  cents  as  often  as 
we  take  1  bushel  at  90  cents. 

It  is  also  evident  that  any  number  of  times  these  proportions  may  be 
taken. 

Hence,  the  different  qualities  of  corn  may  be  taken  in  the  proportions 
of  tV,  \,  i  and  ^;  1,  5,  2  and  1  ;  etc. 

26.  In  what  proportions  may  coffees,  at  85,  40,  50  and  55 
cents  a  pound,  be  mixed  to  produce  a  quantity  worth  45  cents 
per  pound  ? 

Ans.  In  the  proportions  of  1  lb.  at  35  c,  2  lb.  at 
40  c,  2  lb.  at  50  c,  and  1  lb.  at  55  c. 

27.  How  much  gold,  at  20,  21  and  23  carats  fine,  must  be 
mixed  with  12  ounces  20  carats  fine,  so  that  the  mixture  may 
be  22  carats  fine? 

Solution. 

At  2 Oca.  to  gain  lea.  take -oz.;    .     .     .    j  oz.X  24"=  12 oz. 
"  21  ca.       "       lea.    "     loz.;    .     .     .    1  oz.X  24  =  24  oz. 
"  23  ca.  to  lose  lca+  1  ca.  take  loz+  1  oz. ;2oz.X24'^4^ 02. 
12  oz.  ^joz.  =  24  oz. 


PROBLEMS  FOR  ANALYSIS. 


319 


We  find  the  proportions,  without  regard  to  any  of  the  quantities  being 
limited,  to  be  \  oz.  at  20  carats  fine,  1  oz.  at  21  carats,  and  2  oz.  at  23 
carats. 

But  of  the  20  carats  fine  it  is  required  to  take  12  oz.,  or  24  times  \  oz. ; 
hence  the  other  proportions  must  also  be  taken  24  times  as  large.  We 
must  then  take  for  the  required  mixture  12  oz.  at  20  carats  fine,  24  oz.  at 
21  carats,  and  48  oz.  at  23  carats. 

28.  A  grocer  mixed  20  pounds  of  sugar,  worth  15  cents  a 
pound,  with  others  at  16,  18  and  22  cents.  How  many- 
pounds  of  each  were  taken  to  make  a  mixture  worth  17  cents 
per  pound  ? 

Am.  20  lb.  at  15  c,  41b.  at  16  c,  4  lb.  at  18  c., 
8  lb.  at  22  c. 

29.  A  farmer  has  oats  worth  46,  48,  51  and  54  cents  a 
bushel.  What  quantity  of  each  of  these  kinds  must  bo  taken 
to  make  8  bushels  worth  50  cents  a  bushel  ? 


Solution. 

At  ^6  c.  to  gain  1  c.  take  -  hu. 

4 

"  48c.  ''  Ic. 
"  61  c.  to  lose  1  c. 
"  64  c.      "       ie. 


2 

1  hu. 
-  hu. 


-hu.  X  .i  =  i  hu. 

4 

I  hu.  X4  =  2hu. 

Ihu.X  4  =  4  ^^'" 
-  hu.  X  4  =  1  hu. 


2hu. 
S  hu.  -^  2  hu.  =  4- 


8bu. 


We  find  the  proportions,  without  regard  to  the  total  of  the  quantities 
being  limited,  to  be  \  bu.  at  46  cents,  h  bu.  at  48  cents,  1  bu.  at  51  cents, 
and  \  bu.  at  54  cents. 

The  sum  of  these  quantities  is  2  bushels ;  but  the  total  of  the  mixture 
must  be  8  bushels,  or  4  times  as  large;  hence,  each  of  the  quantities 
found  must  be  taken  4  times  as  large.  We,  therefore,  must  take  for  the 
required  mixture,  1  bu.  at  46  cents,  2  bu.  at  48  cents,  1  bu.  at  51  cents, 
and  1  bu.  at  54  cents. 

30.  What  quantities  of  sugars,  worth  3.12,  %.\\  and  ^.08 
per  pound  respectively,  must  be  taken  to  form  a  mixture  con- 
taining 35  pounds,  at  9  cents  per  pound  ? 


320  PEOBLEMS  FOR  ANALYSIS. 

31.  A  grocer  requires  a  cliest  of  tea  containing  75  pounds, 
worth  66  cents  per  pound.  What  quantities  of  several  kinds, 
worth  42, 48, 72  and  78  cents  per  pound,  must  he  mix  to  form  it? 

Am.  9  lb.  at  42  cents,  12  lb.  at  48  cents,  36  lb.  at 
72  cents,  and  18  lb.  at  78  cents. 

Note.— The  last  seven  problems  are  examples  in  what  is  called  AUigatim  Alternate, 
which  is  the  process  of  finding  the  proportions  of  several  articles  of  different  values  that 
may  form  a  quantity  of  a  given  average  value. 

32.  Express  as  a  series  .135135  +,  in  which  the  figures  135 
continually  repeat  in  the  same  order. 

Solution.— The  decimal  .135135  +,  or  .135  .  .  .  ,  may  be  regarded  as 
a  geometrical  progression,  in  which  the  rate  is  ^oV?.  Marking  the  re- 
peating figures  by  placing  a  dot  over  the  first  and  last  of  the  set,  and  we 
have  .135  =  ^^-^-^  +  xirfof ^t  + ,  the  series  required. 

Note.— A  decimal  in  which  a  figure  or  a  set  of  figures  is  repeated  in  the  same  order 
indefinitely  is  called  a  Circulate,  and  the  figure  or  set  of  figures  repeated  is  called  a  Re- 
petend. 

83.  Express  as  a  common  fraction  in  its  lowest  terms  the 
circulate  .27. 

Solution. 

100  times  the  circulate  .27  =  ^7.27 

1  time     "         "  .27 ...  . 


99  times   "         "  =27. 

Hence,  once  the  circulate,  or  -27  =  —-  =  —,  the  fraction  required. 

That  is,  a  repetend  is  equal  to  a  common  fraction  having  for  its  de- 
nominator a-s  many  nines  as  there  are  figures  in  the  repetend,  and  for 
the  numerator  the  figures  of  the  repetend. 

34.  Express  as  a  mixed  decimal  .2259.  Aiis.  .2||-|. 

35.  Express  as  equivalent  common  fractions  .7,  .90  and  .702. 

36.  Express  as  common  fractions  in  their  lowest  terms 
7.936,  8.936  and  32.715.  A,n    «&i    "Rs    T-sni 

37.  E.>ipress  as  conimou  fractions  in  their  lowest "  terms 
.074,  .8145  and  ,138.  ^^.  ^_  ^,^^_  ^, 


GENERAL  REVIEW.  321 

SECTION    LXXVI. 

GENERAL   REVIEW. 

792. — 1.  Add  tliirty-fivc  milliou  eight  hundred  forty  thou- 
sand three  hundred  fifty  ten-thousandths,  four  hundred  sixty- 
three  thousand  nine  hundred  and  eight-hundredths,  and  three 
hundred  four  thousands  and  three  hundred  four  millionths. 

2.  What  number  divided  by  417  will  give  the  quotient  105 
and  the  remainder  113? 

3.  If  j^  of  a  ton  of  hay  cost  $18.50,  how  much  will  tAvo 
loads  cost,  one  weighing  |-  of  a  ton  and  the  other  ^  of  a  ton  ? 

Am.  $27.75. 

4.  20004  +  (20.104  X  5.07)  —  (6.44  --  .0005)  =  what? 

5.  Which  is  the  greater — a  garden  40  rods  square,  or  one 
containing  40  square  rods,  and  how  much  ? 

6.  From  95  mi.  subtract  57  mi.  192  rd.  4  yd.  3  ft.  18  in. 

2 

7.  What  part  of  2|  is  f  of  |  of  f  ?  Ans.  |. 

8.  Find  the  difierence  between  7  thousand  and  7  thou- 
sandths, and  divide  the  remainder  by  7  millionths. 

9.  What  is  the  value  in  compound  numbers  of  .3945  of  a 
day?  Ans.  9  h.  28  m.  4.8  sec. 

10.  How  much  greater  is  the  quotient  of  f  -^-  f  than  the 
product  of  I  X  I? 

11.  A  hall,  50  feet  long  and  30  feet  wide,  has  around  it  a 
mop-board  9  inches  high.  The  hall  has  one  door  6  feet  wide 
and  2  doors  3  feet  wide.  How  many  square  feet  are  there  in 
the  surface  of  the  mop-board  ? 

12.  If  a  merchantman,  sailing  9|-  knots  an  hour,  is  chased 
by  a  gun-boat  steaming  lOf  knots,  how  far  ahead  must  the 
sailing  vessel  be  to  escape  3  knots  ahead  into  a  port  from  which 
she  is  15|-  knots  at  the  commencement  of  the  chase? 

13.  If  by  selling  a  horse  at  $80  I  lose  12^%  of  the  first  cost, 
shall  I  gain  or  lose,  and  what  per  cent.,  by  selling  him  at  $90  ? 

14.  How  many  times  has  February  29th  occurred  since  the 
year  1799? 


322  GENERAL    REVIEW. 

15.  A  merchant  bought  f  of  a  hogshead  of  molasses  at  f  of 
a  dollar  a  gallon.  At  what  price  per  gallon  must  he  sell  it  to 
make  §4.20  ?  Ans.  $.70. 

16.  I  bought  apples  at  $5  per  barrel,  and  lost  one  fourth  of 
them.  At  what  price  must  I  sell  the  rest  that  I  may  gain 
10%  on  the  whole  cost?  Ans.  ^1.Z^. 

17.  If  6  men  in  8  hours  thresh  30  bushels  of  wheat,  in  how 
many  hours  can  5  men  thresh  50  bushels  ? 

18.  A  has  a  farm  |-  of  a  mile  square,  and  B  has  one  contain- 
ing y^g-  of  a  square  mile.     How  do  the  farms  compare  in  size  ? 

19.  An  agent  received  867.50  for  collecting  84500.  What 
was  the  rate  of  his  commission  ? 

20.  What  is  the  least  number  which,  being  divided  by  3,  by 
5,  by  7,  by  9  and  by  10,  leaves  in  each  case  a  remainder  of  2? 

21.  How  much  shall  I  gain  by  borrowing  $3560  for  1  year 
6  months  10  days,  at  6%,  and  lending  it  at  7%  ? 

22.  I  obtained  a  discount  at  a  bank  at  7%,  and  left  \  of  the 
proceeds  in  the  bank  until  the  note  was  paid.  At  what  rate 
did  I  get  the  money  I  used  ? 

23.  If  14  men  can  perform  a  piece  of  work  in  36  days,  in 
how  many  days  can  they  perform  the  same  labor  with  the 
assistance  of  7  more  men  ? 

24.  How  much  more  time  will  it  require  a  sum  of  money  to 
double  itself  at  6  %  interest  than  at  7  %  ? 

25.  AVhat  is  the  amount  of  $1450.40  from  April  19,  1871, 
to  August  3,  1872,  at  6%  ? 

26.  What  is  the  difference  between  the  simple  and  the  com- 
pound interest  on  $5000  for  ^  years,  at  7  %  ?   Ans.  $114.60. 

27.  A  man  bequeathed  \  of  his  estate  to  his  wife,  ^  to  a 
college  and  ^  to  his  eldest  son;  and  these  three  legacies 
amounted  to  $18500.     How  much  did  each  receive? 

Ans.  Wife,  $7500  ;  college,  $6000 ;  son,  $5000. 

28.  A  merchant  paid  $4200  for  cotton,  and  sold  it  at  10% 
advance,  taking  his  })ay  in  prints,  which  he  sold  at  a  loss  of 
10%.     Did  he  gain  or  lose,  and  how  much? 

29.  What  is  the  present  worth  of  $770,  due  in  1  year  8 
mouths,  at  G%  ?  Ans.  $700. 


GENERAL    REVIEW.  323 

30.  A  and  B  traded  in  company.     A  put  in  $950,  and  B 
They  gained  $300.     What  was  each  partner's  share  of 

it?  Ans.  A's,  S162f ;  B's,  $137|. 

31.  A  person  has  a  field  measuring  3  acres  75  square  rods, 
•which  he  wishes  to  exchange  for  a  square  one  of  inferior 
quality,  but  3|-  times  as  large.  How  many  rods  is  the  length 
of  its  side  ?  Ans.  44.0738. 

32.  Smith  and  Doland  trade  in  company,  Smith  contributing 
$800  for  9  months,  and  Doland  $600  for  8  months.  They 
gain  $450.     What  should  each  receive  ? 

33.  What  is  the  area  of  a  triangle  whose  base  is  b\  yards, 
and  whose  altitude  is  8^  yards  ?  Ans.  23f  sq.  yd. 

34.  Gold  is  soiling  at  112.  Find  the  interest  in  currency  on 
7  $1000  U.  S.  5-20  bonds  from  June  1,  1870,  to  July  1, 
1871. 

35.  A  person  being  asked  the  time  of  day,  replied  that  it 
was  between  5  and  6  o'clock,  and  that  the  hour-  and  minute- 
hands  of  the  watch  Avere  together.     What  was  the  time  ? 

Ans.  ^1-Yi  minutes  past  5  o'clock. 

36.  If  the  diameter  of  a  9-pound  cannon-ball  be  5  inches, 
what  must  be  the  diameter  of  a  28-pound  ball?    Ans.  7.3  in. 

37.  A  merchant  bought  goods  to  the  amount  of  $1400,  on  a 
credit  of  6  months.  At  the  end  of  3  months  he  paid  $600, 
and  1  month  later  $400.  What  extension  ought  he  to  have  on 
the  balance  of  the  debt?  Ans.  6|-  months. 

88.  Wishing  to  find  the  distance  between  two  trees,  wliich 
cannot  be  directly  measured  on  account  of  an  intervening 
pond,  I  measure  due  west  50  rods  from  the  foot  of  one  of 
them ;  tlien,  turning  north,  measure  34  rods,  when  I  find  that 
I  am  just  20  rods  west  of  the  other  tree.  How  far  are  the 
trees  apart  ?  Ans.  45.34  +  rods. 

39.  A  gentleman  selling  a  mortgage  of  $4410,  for  which  ho 
received  5%  interest,  invested  the  proceeds  in  Government 
8^  %  bonds  at  70.  After  receiving  the  interest  for  5  years,  on 
the  bonds  rising  to  75,  he  sold  out.  What  was  his  gain  upon 
the  Avhole  transaction  over  what  he  would  have  received  had 
he  continued  the  mortgage  ?  Ans.  $315. 


324  GENERAL  REVIEW. 

40.  After  the  outbreak  of  the  Prusso-French  war  in  1870, 
the  Prussian  Government  issued  a  5%  war  loan  at  88.  The 
French  3  per  cents,  stood  at  65|.  State  the  ratio  of  the  two 
rates  of  interest.^  Am.  6||  to  4^V 

41.  A  joist  is  7-|^  inches  wide  and  2^  thick,  but  I  want  one 
just  twice  as  large,  which  shall  be  3f  inches  thick.  What 
must  be  the  width  ?  Ans.  10  inches. 

42.  A  and  B  can  do  a  piece  of  work  alone  in  12  and  16 
days  respectively.  They  labor  together  on  the  work  for  3 
days,  when  A  leaves  it,  but  B  continues,  and  after  2  days  is 
joined  by  C,  and  they  finish  it  together  in  3  days.  In  what 
time  would  C  do  it  alone  ? 

43.  How  many  cubic  yards  of  gravel  will  be  required  for  a 
■walk  surrounding  a  rectangular  lawn  200  yards  long  and  100 
yards  wide,  the  walk  to  be  3  yards  wide,  and  the  gravel  3 
inches  deep  ? 

44.  A  saves  \  of  his  income;  but  B,  who  has  the  same 
income,  spends  twice  as  fast  as  A,  and  thereby  contracts  a  debt 

'  of  $120  annually.     What  is  the  income  of  each  ? 

Ans.  $360. 

45.  A  stole  a  horse  from  B,  and  made  off  with  him.  Five 
days  afterward,  B  gets  intelligence  of  A,  and  follows  him  at 
the  rate  of  60  miles  a  day,  by  which  he  gains  20%  upon  A. 
How  far  must  B  ride  to  overtake  A,  and  how  many  days  ? 

Ans.  1500  miles ;  25  days. 

46.  A  steamer,  working  with  a  given  force,  can  run  down 
the  river  at  the  rate  of  12|  miles  per  hour.  Of  this  speed,  f 
is  due  to  the  current.  How  long  would  the  steamer  require  to 
go  15  miles  up  the  stream  ? 

47o  Subtract  the  square  root  of  ^^4^  f^'om  the  cube  root  of 
the  same.  Ans.  -j. 

48.  The  plan  of  a  town  is  11  \  inches  long  and  14  inches 
broad,  and  the  scale  annexed  to  it  is  jui^t  2|-  inches  to  1100 
>ards.  What  is  the  length  of  a  mile  upon  this  scale,  and 
what  will  be  the  length  and  breadth  of  the  pUxn  if  it  bo 
enlarged  to  a  scale  of  6  inclics  to  a  mile  ? 

Ans.  4|  inches  ;  25  inches  by  20  inches. 


APPENDIX. 


ROMAJ^'  JfOTATIOM. 

793.  Roman  Notation  uses  seven  letters :  I,  V,  X,  L,  C,  D 

find  M,  which  express,  respectively,  one,  five,  ten,fijtij,  one  hun- 
dred, five  hundred  and  one  thousand. 

All  numbers  can  be  expressed  by  these  letters,  used  singly, 
or  combined  according  to  the  following 

794.  Principles. — 1.  When  a  letter  is  repeated,  the  number 
ivhich  it  expresses  is  repeated. 

Thus,  11  =  1  +  1  =  2;  XXX  =  10  +  10  +  10  =  30;  CC  =  100  + 
100  =  200. 

2.  When  a  letter  e.rpressing  a  certain  number  stands  after  one 
expressing  a  greater  number,  the  sum  of  the  numbers  is  denoted. 

Thus,  VI  =  5  +  1  =  6 ;  XI  =  10  +  1  =  11 ;  LX  =  50  +  10  =  60. 

3.  When  a  letter  expressing  a  certain  number  stands  before 
one  expressing  a  greater  number,  the  difference  of  the  numbers  is 
denoted. 

Thus,  IV  =  5  —  1  =  4 ;  IX  =  10  —  1  =  9 ;  XL  =  50  —  10  =  40. 

4.  JVlien  a  letter  expressing  a  certain  number  stands  between 
two  letters  expressing  greater  numbers,  the  least  number  is  to  be 
subtracted  from  the  sum  of  the  other  tivo. 

Thus,  XIV  =  (10  +  5)  —  1  =  14 ;  CXL  =  (100  +  50)  —  10  =  140. 

5.  A  bar,  — ,  placed  over  a  letter  makes  it  denote  thousands. 
Thus,  V  =  5000 ;   D  =  500,000  ;  M  =  1,000,000. 

EXERCISES. 

Write  and  read — 

1.  XIX.  3.  LXXXV.        5.  MDCCCLXXV. 

2.  CXVI.         4.  DCLXX.         6.  MLIXCXX. 

Express  by  letters — 

7.  Forty-eight.  9.  One  thousand  six  hundred  eleven. 

8.  Two  hundred  five.       10.  Eighteen  hundred  seventy-nine. 

28  325 


S26  •  CONTRACTIONS. 

COJs'TRACTIOJs^S. 
To  Add  Two  Coiumus  at  a  Time. 

795.— Ex.  1.  Add,  two  columns  at  a  time,  1235,  6714,  4566 

and  4967. 
1235        Solution.— 67  +  6  =  73,  +  60  =  133,  +  4  =  137,  +  10 
G71A     ""  ^^'^'  +  o  =  152,  +  30  =  182  ;   write  82.     1  +  49  =  50, 
/  nrr     +  5  =  55,  +  40  =  95,    +7  =  1  2,  +  60  =  162,  +  2  = 
40 0  0      ig4^  +  10  =  174;  write  174.  Ans.  17482. 

4967         In  practice;  thus,  67,  73,  133,  137,  147,  152,  182;  write  82. 
17482     ^'  ^^'  ^•^'  ^5,  102,  162,  164,  174;  write  174.        Ans.  17482. 

796.  Rule  for  Adding  two  Columns  at  a  Time.— Jb  the  low- 
est niunher  add  the  ones  of  the  next  ninnher  above, 
then  add  the  tens  of  that  mnribev ;  to  the  sinyv  thus 
obtained  add  the  ones  of  the  next  number  above, 
then  the  tens  of  that  number,  and  so  on. 


JPJtOBrEMS. 

(1.) 

(2.) 

(3.) 

(4.) 

(5.) 

36 

4402 

6645 

47 

4141 

71 

6307 

5232 

81 

3226 

58 

1453 

7070 

92 

1819 

32 

9205 

3007 

31 

4234 

43 

1824 

7084 

35 

1781 

64 

7132 

2636 

74 

9603 

50 

1042 

2273 

60 

2009 

To  Multiply  by  a  Multiplier  of  Two  Orders  at  Once. 
797.— Ex.  1.  Multiply  1246  by  32. 

1246         Solution— 6  X  32  =  192;    write  2.     4  X  32  =  128, 
32     +  19  =  147  ;  write  7.     2  X  32  =  64,  +  14  =  78 ;  write  8. 


39872     1  X  32  =  32,  +  7  =  39  ;  write  39.  Ans.  39872. 

798.  Rule  for  Multiplying  by  a  Multiplier  of  Two  Orders  at  Once.— 
Mnlii])ly  each  order  of  the  multiplicand  separately 
by  the  entire  multiplier. 


Multiply — 


CONTRA  CTIONS.  327 

PltOBLEMS. 


1.  7418  by  35 ;  by  42. 

2.  6320  by  15 ;  by  53. 


3.  91367  by  44;  by  61. 

4.  34205  by  67  ;  by  88. 


To  Multiply  by  an  Aliquot  Part  of  10,  100,  etc. 
799.— Ex.  1.  Multiply  3465  by  125. 
8)34-65000         Solution. — Since  125  is  one  eighth  of  1000,  multiply 

ooTTTt"      ^^'  1000,  and  take  one  eighth  of  the  product. 

4.33125  ^„g  433125. 

800.  Rule  for  Multiplying  by  an  Aliquot  Part  of  10,  100,  etc.— 

Multiply  by  10,  100,  etc.,  and  of  the  product  thus 
obtained  take  such  a  part  as  the  given  multiplier 
is  of  the  multiplier  used. 


mOBZEMS. 

Multiply — 


1.  674  by  2| ;  by  250. 

2.  342  by  8^ ;  by  33^. 

3.  758byl2i;  by  125. 


4.  8910  by  16| ;  by  333^. 

5.  7648  by  25  ;  by  250. 

6.  68024  by  331;  by  125. 


To  Divide  by  an  Aliquot  Part  of  10,  100,  etc. 

801._Ex.  1.  Divide  433125  by  125. 

433.125 

Q         Solution. — Since  1000  is  8  times  125,  divide  by  1000, 

and  take  8  times  the  quotient.  Ans.  3465. 


3465.000 


802.  Rule  for   Dividing    by  an  Aliquot  Part  of  10,   100,  etc.— 

Divide  by  10,  100,  etc.,  as  the  problem  may  require, 
and  multiply  the  quotient  thus  obtained  by  the  num- 
ber which  shows  how  lyuiny  times  the  given  divisor 
is  contained  in  the  divisor  used. 


PItOBLEMS. 

Divide — • 


1.  16850  by  25  ;  by  125. 

2.  5700  by  2^ ;  by  84. 

3.  25300  by  12| ;  by  250. 


4.  7360  by  16| ;  by  33|. 

5.  7600  by  250  ;  by  333|. 

6.  552642  by  50 ;  by  125. 


328 


DUODECIMALS. 


DUODECIMALS. 
803.  A  Duodecimal  is  a  deuomiuate  number  in  which  a  unit 
of  any  denomination  is  equivalent  to  twelve  units  of"  the  next 
lower   denomination  ;    or,  it  may  be  regarded  as  a  series  of 
fractions  whose  denominators  are  successive  powers  of  12. 

Note. — Examples  in  DuoJecimals  can  generally  be  more  readily  performed  by  reducing 
the  Duodecimals  to  Common  or  Decimal  Fractions,  but,  since  the  special  rules  are  used  by 
gome  mechanics  in  measuring  surfaces  and  solids,  it  is  thought  best  to  give  them  here. 

In  duodecimals  the  foot  is  taken  as  the  unit ;  twelfths  of  a  foot  are 
called  primes ;  twelfths  of  a  prime,  seconds ;  twelfths  of  a  second,  thirds,  etc. 

Primes  are  marked  ^ ;  seconds,  '^ ;  thirds,  ^^^ ;  fourths,  ^^^^,  etc.,  and 
the  marks  are  called  indices. 


For  Lengths. 

1ft.  =  12'      =  12  in. 

V       =  12"     =    i    " 

1"      =  12'"   =  —  " 
12 


TABLES. 

For  Surfaces. 

lft.=   12'     =lUsq.in. 
1'    ■  =  12"    =    12    " 
1"     =  12'"  =      i    " 
1'"    =  12""  =      L    " 


For  Volumes. 

1ft.  =  12'     =1728  cu.  in. 
1'     =  12"    =  lU     " 
1"    =  13'"  =    12      " 
1'"  -  12""  =     1       " 


ADDITION  AND  SUBTRACTION. 
804.  Duodecimals  are  added  and  subtracted  in  the  same 
manner  as  compound  numbers  are. 

Thus,  23  ft.  %'W'  IV  +  15  ft.  8^  7'^  8'^'  =  39  ft.  6'  6'^  1'"  \ 
12  ft.  3'  1"  W"  %""  —  8  ft.  8^  %"  =  3  ft.  6^  10^^  %'"  W". 


MULTIPLICATION. 
805. — Ex.  1.  How  many  square  feet  in  a  board  12  feet  9 
inches  long  and  2  feet  6  inches  wide? 

9' 


12  ft. 

2 


6' 


6  ft. 

25 


4 

6' 


6" 


31  f 


31ft 

'1  +  , 


10'    6"  = 
-  31.875  sq.ft. 


Solution.  — 9  in.,  or  9^,= 
t\  ft.,  and  6  in.,  or  6^  =  ^%  ft. 
9^X6'=54''  =  4'6''.  Write 
the  &'',  and  add  the  A'  to  the 
next  product.  12  ft.  X  6'  =-- 
72';  72'  +  ¥  =  76'  =  6  ft.  4', 
which  we  write  in  the  product. 
9'X  2  =  18'  =  1  ft.  6'.  Write 
the  6',  and  add  the  1  ft.  to  the 
24  ft.  ^-  1  ft.  3=  25  ft.,  whicli  we 


next  product.      12  ft.  X  2  =^  24  ft 

write  in  the  pro<luct.    Adding  the  partial  products,  we  have  31  ft.  10'  6 


D  UODE  CIMALS.  329 

12  75  ^"*^  reducing  the  primes  and  seconds  to  the  decimal 


2.5 


of  a  foot,  we  liave  31.875  sq.  ft 


Solution  by  Decimals. — Expressing  the  inches 
^^ I  ^  as  decimals  of  a  foot,  and  multiplying  (Art.  398),  we 

2550  obtain,  more  readily  than  by  duodecimals,  the  same 

31,875  sq.  ft.     result  as  before. 

806.  Rule  for  Multiplication  of  Duodecimals.— Multiply  as  iii 

multiplicatiorv  of  integers,  ohserving  that  tivelve  of 
each  duodecimal  deiioininatioih  are  one  of  the  next 
higher. 

"=•  PROBLEMS. 

1.  What  is  the  area  of  a  plank  20  ft.  3  in.  long  and  1  ft. 
8  in.  wide  ?  Ans.  33.75  sq.  ft. 

2.  What  are  the  cubic  contents  of  a  stick  of  squared  tim- 
ber 20  ft.  long,  18  in.  wide  and  14  in.  thick?     Ans.  35  cu.  ft. 

3.  How  many  square  feet  of  flooring  in  a  room  24  ft.  7  in. 
long  and  16  ft.  4  in.  wide?  Ans.  401  sq.  ft.,  76  sq.  in. 

DIVISIOX. 
807— Ex.  1.  Divide  19  ft.  10'  11"  8'"  by  2  ft.  4'. 

19  ft.  l(j  if  8'"  =  3441^'  Solution.— The    divi- 

2  ft.     4  =     4032"'  dend  is  equal  to  34412'^' 

^  ,     „     -  the    divisor    to    4032''^ 

344^"^  -^  4032  =  Sj^  =  oft.  6  O  Dividing,  we  have  a  quo 

tient  8  and  a  remainder  2156.  Reducing  this  remainder  to  twelfths  or 
primes,  and  dividing,  we  have  6^  and  a  remainder  1680'.  Reducing 
this  remainder  to  twelfths  of  primes,  or  to  seconds,  and  dividing,  we 
have  b'',  and  no  remainder.     Hence,  the  entire  quotient  is  8  ft.  6'  b". 

808.  Rule  for  Division  of  Duodecimals.— i^ec^i^ce  the  divi- 
dend and  divisor  to  the  loivest  denomination  found 
in  either,  and  divide  as  in  integers.  The  quotient 
uill  he  an  integer.  If  there  he  a  remainder,  reduce  it 
to  twelfths  or  primes,  and  continue  the  division;  re- 
duce the  second  remainder,  if  any,  to  twelfths  of 
j)j'imes,  or  seconds^  and  proceed  as  hefore. 

PROBLEMS. 

1.  Divide  90  ft.  3'  6"  by  5  ft.  6'.  Ans.  16  ft.  5'. 

2.  Divide  55  ft.  9'  0"  10'"  by  5  ft.  7'.     Ans.  9  ft.  11'  10". 
28* 


330  ACCURATE  INTEREST. 

ACCURATE  INTEREST. 

809.  The  United  States  Government  pays  Accurate  Interest, 

reckoning  365  days  to  the  year. 

Usual  interest  makes  each  day's  interest  -g^  of  a  j^ear's 
interest ;  and  to  equal  exact  interest  must  be  diminished  by 
^^  of  itself  in  a  common  year,  or  by  ^  of  itself  in  a  leap 
year. 

810. — Ex.  1.  What  is  the  interest  on  a  Government  note  of 
$1000  for  sixty  days,  at  5%  ? 

$50y.G0         (ft  Solution. — Interest  on  $1000  for  one  year 

— — —  =  $8.22.      =  $50.   Interest  for  60  days,  or  ^^V  of  a  year, 

"^^^  §2£J1J9  =  $8.219 +.  J.71S.  $8.22. 

365 

811.  Rule  for  Accurate  Merest— MuItiplT/  the  iivterest 
for  one  year  at  the  given  rate  hy  the  given  nwmber 
of  days,  and  divide  the  product  hy  365. 

FROBT.JEMS. 

1.  What  is  the  interest  on  a  Government  bond  of  $500  for 
31  days,  at  6%? 

2.  What  is  the  interest  on  a  ten-forty  U.  S.  bond  of  $5000 
from  Juno  11  to  August  21  ? 

3.  What  is  the  interest  on  U.  S.  securities  of  S18000  from 
April  4  to  July  13,  at  ^%  ?  Ans.  $221.92. 


TEST   QUESTIONS. 

812. — 1.  What  characters  are  used  in  expressing  numbers  byKoMAN 
Notation?     What  are  the  principles  of  Koman  notation  ? 

2.  What  is  the  Rule  for  adding  two  cohimns  at  a  time?  For  multi- 
plying bv  a  multiplier  of  two  orders  at  once?  For  nniltiplying  by  an 
aliquot  p.-vrt  of  10,  100,  etc.?  For  dividing  by  an  aliquot  part  of  10, 
100,  etc.? 

3.  Wliat  is  a  Duodecimal  ?  How  are  duodecimals  added  and  sub- 
tracted ?  What  is  the  rule  for  multiplication  of  duodecimals?  For 
division  of  duodecijuals  ? 

4.  What  is  Acouratk  Intehkst?  What  does  usual  interest  make 
each  day's  hiterest  ?     What  is  tlie  rule  for  accurate  interest  ? 


EXAMI2\^ATI0N  PROBLEMS.  331 

.  EXAMIJ^ATIOK  PROBLEMS. 

813,  The  following  Problems  may  be  used  at  the  discretion 
of  the  teacher  in  testing  the  proficiency  of  pupils  as  they 
progress  in  the  book.  The  Articles  in  parentheses  denote  the 
portions  of  the  text  to  which  the  problems  relate. 

(Articles  1—139.) 

1.  Express  in  words  6115789023665724. 

2.  Represent  by  figures  fifteen  quadrillions  four  hundred 
one  trillions  eleven  millions  seventeen. 

3.  Show  by  an  example  the  use  of  0  in  writing  numbers. 

4.  A  has  795  dollars,  B  has  105  more  than  A,  and  C  has  as 
many  as  A  and  B.     How  many  dollars  have  they  all  ? 

5.  From  one  million  take  five  hundred  thousand  five. 

6.  A  man  purchased  a  house  for  7500  dollars ;  after  paying 
560  dollars  for  repairs,  and  receiving  475  dollars  for  rent,  he 
sold  it  for  8000  dollars.     How  much  did  he  gain  ? 

7.  The  difierence  between  two  numbers  is  1162,  and  the 
larger  number  is  9340.     What  is  the  smaller  ? 

8.  Illustrate  by  an  example  a  method  of  proving  results  in 
Addition. 

9.  How  many  men  are  there  in  an  army  of  112  regiments, 
each  of  which  consists  of  947  men  ? 

10.  I  bought  35  cows  at  52  dollars  a  head,  and  27  at  60  dol- 
lars a  head,  and  sold  the  whole  at  54  dollars  a  head.  How 
much  did  I  gain  or  lose  ? 

11.  The  product  of  two  factors  is  224638568,  and  one  of  them 
is  729346.     What  is  the  other  ? 

12.  The  quotient  of  the  exact  division  of  one  number  by 
another  is  3168.  What  would  have  been  the  quotient  if  the 
divisor  had  been  6  times  as  large? 

13.  Show  by  an  example  that  Division  is  the  reverse  of  Mul- 
tiplication, 

14.  What  is  the  value  of  (8+16  x  6j  ^  3  —  ^§2^d^)>il? 


332  EXAMIXATION  PROBLEMS. 

15.  How  many  tons  of  hay  at  32  dollars  a  ton,  can  be  ex- 
changed for  44  tons  of  coal  at  8  dollars  per  ton  ? 

16.  Which  of  the  numbers  84,  282  and  798  is  divisible  by 
the  largest  prime  number  ? 

17.  How  much  less  is  the  greatest  common  divisor  of  30  and 
42  than  their  least  common  multiple  ? 

18.  What  is  the  smallest  sum  of  money  with  which  I  can 
purchase  colts  at  25  dollars  each,  cows  at  40  dollars  each,  oxen 
at  100  dollars  each,  or  horses  at  125  dollars  each? 

(Articles  140— S96.) 

19.  How  much  does  f  +  f  exceed  f  —  f? 

20.  The  sum  of  two  fractions  is  |-|,  and  one  of  them  is  f . 
What  is  the  other  ? 

21.  What  number  divided  by  3|-  will  give  ^? 

22.  If  f  of  a  yard  of  cloth  costs  |-^  of  a  dollar,  what  will  be 
the  cost  of  1  yard  ? 

23.  Show  by  examples  the  difference  between  a  Compound 
Fraction  and  a  Complex  Fraction. 

24.  What  is  ^\  of  a  ship  worth  if  \^  of  it  be  worth  50000 
dollars  ? 

25.  A  man  spent  f  of  -g-  of  his  money  one  day,  f  of  |-  of  it 
another  day,  and  then  had  3887  dollars  remaining.  How 
much  money  had  he  at  first? 

26.  Show  by  examples  that  multiplying  or  dividing  both 
terms  of  a  fraction  by  the  same  number  does  not  change  the 
value  of  the  fraction. 

27.  What  common  fraction  is  equal  to  the  simi  of  .655,  .33^ 
and  .9375  ? 

28.  Express  by  figures  six  hundred  thousand  six,  and  six 
million  sixty  thousand  six  hundred  six  billionths.    , 

29.  How  much  less  than  one  million  is  one  millionth  ? 

30.  A  63-gallon  cask  is  f  full  of  wine ;  if  27.625  gallons 
should  leak  out,  the  wine  remaining  will  be  what  decimal  part 
of  a  full  cask  ? 

31.  Show  by  an  example  how  Integers  and  Decimals  cor- 
respond in  expression. 


EXAMINATION  PROBLEMS.  333 

32.  What  will  56.75  acres  of  land  cost  at  $20.25  per  acre  ? 

33.  What  is  the  value  of  ^^^  +  II_ ^^^^  v 

.005        4        5.5 

34.  A  grocer  paid  $58G. 50  for  apples,  giving  S2.25  a  barrel 
for  124  barrels,  and  $3.75  for  the  remainder.  How  many  bar- 
rels did  he  buy  ? 

35.  How  many  times  is  .029  exactly  contained  in  .3786,  and 
what  will  remain  ? 

36.  A  sum  of  money  was  divided  among  three  boys  ;  the  first 
received  .375  of  the  whole ;  the  second,  .6  of  the  whole ;  and 
the  third,  ^2.12|-.     What  was  the  sum  divided  ? 

(Articles  297—407.) 

37.  How  many  inches  are  there  in  1051  yards  2  feet  5  inches  ? 

38.  Reduce  3186938  seconds  to  days. 

39.  What  will  28  square  rods  129  square  feet  of  land  cost, 
at  12  cents  per  square  foot? 

40.  In  walking  from  one  town  to  another  a  man  took  29700 
steps  of  2  feet  8  inches  each.     How  many  miles  did  he  walk? 

41.  Show  by  an  example  that  Reduction  Descending  and 
Reduction  Ascending  are  reverse  processes. 

42.  What  is  the  sum  and  the  difference  of  75  yards  1  foot 
9  inches  and  46  yards  2  feet  11  inches? 

43.  On  July  17,  1872,  how  old  was  a  man  Avho  was  born 
February  18,  1819? 

44.  How^  many  cords  are  there  in  3  ranges  of  wood,  each  being 
12  feet  long,  4  feet  w'ide  and  6  feet  4  inches  high  ? 

45.  What  is  the  product  of  16°  15'  16"  multiplied  by  11  ? 

46.  How  much  must  be  paid  for  41  gal.  2  qt.  If  pt.  of 
molasses,  at  72  cents  a  gallon  ? 

47.  What  part  of  a  cubic  yard  is  a  cube  whose  edge  is  one- 
half  of  a  yard  ? 

48.  What  fraction  of  an  ounce  Troy  is  15  pwt.  9y\  gr.  ? 

49.  How  much  must  be  paid  for  one-seventieth  of  336  bu. 
3  pk.  4  qt.  of  corn,  at  SO  cents  a  bushel  ? 

50.  How  many  cubic  feet  in  a  rectangular  beam  24  feet 
6  inches  long,  1  foot  9  inches  wide  and  1  foot  2\  inches  thick  ? 


334  EXAMINATION  PROBLEMS. 

51.  What  decimal  of  a  ton  is  f  of  an  ounce? 

52.  How  much  hay,  at  $30  a  ton,  can  be  bought  for  8131.25  ? 

53.  Express  ^  oi  a.  day  in  hours,  minutes  and  seconds. 

54.  From  a  piece  of  land  24  rods  square  I  sold  -^  of  an  acre 
to  A,  f  of  an  acre  to  B,  and  .675  of  an  acre  to  C.  How  much 
of  the  land  was  left  ? 

55.  A  ship's  chronometer,  set  at  Philadelj)hia,  longitude 
75°  10'  W.,  pointed  to  3  h.  40  min.  24  sec.  a.m.  when  the  sun 
was  on  the  meridian.     In  what  longitude  was  the  ship  ? 

(Articles  408—531.) 

56.  If  I  sell  land  at  S75  per  acre,  and  thereby  gain  25%, 
how  much  per  acre  did  the  land  cost  me  ? 

57.  How  much  is  10%  of  25%  of  1680  bushels? 

58.  I  bought  a  horse  for  $120,  and  sold  him  for  8160. 
What  per  cent,  did  I  gain  ? 

59.  My  house  is  worth  y%  as  much  as  my  farm.  What  per 
cent,  of  the  value  of  the  farm  is  the  value  of  the  house  ? 

60.  A  received  from  B  85100,  with  which  he  bought  flour 
at  85  per  barrel,  deducting  his  commission  of  2%  on  the  cost. 
How  many  barrels  did  he  buy  ? 

61.  What  are  the  interest  and  the  amount  of  88500  for 
2  years  7  months  21  days,  at  7%  ? 

62.  William  bought  cows  at  880  each,  and  one-fifth  of  them 
died.  At  what  price  must  he  sell  the  rest,  to  gain  5%  on  the 
whole  ? 

63.  How  long  must  8600  remain  on  interest  at  6%  to  gain 

81408? 

64.  What  is  the  interest  of  812750  for  5  years  5  months  18 
days,  at  6%? 

<6o.  What  must  be  the  face  of  a  note  payable  in  90 
(lays,  on  which  85000  would  be  received  from  a  bank,  dis- 
counting at  5%  ? 

66.  How  much  will  the  com])ound  interest  exceed  the  annual 
interest  on  81000  for  3  years  6  months,  at  6%  ? 

67.  What  is  the  difference  between  the  true  and  the  bank 
discount  of  8500  for  3  months,  at  8%  ? 


EXAMINATION  PROBLEMS.  335 

68.  What  principal  on  interest  at  7%,  from  April  9,  1871, 
to  September  5,  1873,  will  amount  to  $1477.59  ? 

69.  The  difference  between  the  interest  of  $600  and  that  of 
$750,  at  b<'/f,  for  a  certain  time,  is  $18.75.     "What  is  the  time? 

70.  What  sum,  paid  May  16,  will  settle  a  bill  of  $850.50 
for  goods  bought  April  19,  on  60  days'  credit,  the  rate  of  in- 
terest being  7  %  ? 

71.  A  note  for  $1740,  dated  June  15,  1869,  with  interest  at 
69^,  was  indorsed  as  follows  :  Jan.  1, 1870,  received  $100  ;  July 
15, 1870,  received  $112  ;  June  1, 1871,  received  $200  ;  and  May 
1,  1872,  received  $600.     What  was  due  Sept.  1,  1872? 

(Arlieies  535 -«51.) 

72.  Two  numbers  are  108  and  27 ;  what  is  the  ratio  of  the 
second  to  the  first  ? 

78.  Two  numbers  whose  sum  is  3410  are  in  the  ratio  of  5 
to  6.     What  are  the  nund^ers  ? 

74.  How  many  men  can  jierform  a  piece  of  work  in  112  days 
which  12  men  can  perform  in  84  days? 

75.  If  18  men  can  dig  a  trench  30  yards  long  in  24  days 
by  working  8  hours  a  day,  how  many  men  can  dig  a  trench  60 
yards  long  in  64  days,  working  6  hours  a  day  ? 

76.  Show  by  examples  the  difference  between  Simple  and 
Compound  Proportion. 

77.  If  A  invests  in  a  certain  enterprise  $600  for  4  months, 
B  $300  for  7  months,  and  C  $200  for  9  months,  what  part  of 
the  profits  should  each  of  them  have  ? 

78.  A,  B  and  C  are  partners.  A  furnishes  \  of  the  capital ; 
B,  $500  ;  and  C,  $400.  At  the  end  of  the  year  the  profits  are 
$4200.     What  sum  should  each  receive  ? 

79.  I  owe  Jones  two  bills,  one  of  $600  on  a  credit  of  60  days, 
and  the  other  of  $800  on  a  credit  of  30  days.  These  bills 
being  without  grace,  in  how  many  days  should  a  note  given  for 
their  amount  be  made  payable  ? 

80.  The  balance  of  an  account  is  $420,  and  is  due,  by  aver- 
age, April  28.  What  was  its  cash  value  March  8,  interest  being 
at8%? 


336  EXAMINATION  PROBLEMS. 

81.  HoAV  much  must  be  invested  in  Government  4  per  cents., 
at  93f,  to  realize  a  quarterly  interest  of  S30? 

82.  What  will  be  the  cost  of  a  draft  of  $12500,  at  60  days, 
exchange  being  at  100|-,  and  interest  at  7%  ? 

83.  What  rate  is  paid  for  money  when  l\^o  is  charged  for 
exchange  on  a  30-day  note  discounted  at  6%  ? 

84.  I  invested  $1606  in  4J-  per  cents,  at  lOOi,  and  sold  when 
they  had  fallen,  losing  $100,  inclusive  of  the  double  brokerage 
of  -1%.     At  what  price  did  I  sell? 

(Avtieles  652—792.) 

85.  What  integral  power  of  5  is  nearer  than  any  other  to 
100000? 

86.  Show  by  an  example  the  difference  between  the  Square 
Root  and  the  Cube  Root  of  a  number. 

87.  What  is  the  difference,  carried  to  3  orders  of  decimals, 
between  1^3  and  l/2  ? 

88.  What  difference  is  there  between  the  area  of  a  floor  30 
feet  square  and  that  of  three  others  each  10  feet  square? 

89.  Two  ships,  A  and  B,  sailed  from  a  certain  port  at  the 
same  time.  A  sailed  north,  8  miles  an  hour,  and  B  sailed  east, 
6  miles  an  hour.  How  far  apart  were  they  at  the  end  of  the 
third  hour  ? 

90.  What  is  the  product  of  7.3  multiplied  by  1.92? 

91.  Show  by  examples  the  difference  between  Arithmetical 
and  Geometrical  Progression. 

92.  A  room  is  20  feet  long,  16  wide  and  10  feet  high.  What 
is  the  distance  from  an  \ipper  corner  to  the  opposite  lower 
corner  ? 

93.  How  many  feet,  board  measure,  in  a  rectangular  beam 
14  feet  9  inches  long,  1  foot  8  inches  wide  and  1  foot  4  inches 
thick? 

94.  A  garden  is  100  feet  long  and  80  feet  wide.  What  must 
be  the  dimensions  of  a  similar  garden  to  contain  just  one-half 
as  many  square  feet  ? 


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Royse's  Manual  of  American  Literature. 

Thk  Plan  of  this  Work  may  be  Ijricfly  stated  as  follows: — 
First. — It   presents  a  succinct  historical  retrospedt  or  resume  of  the  origin 
and  progress  of  American  Literature ;  noticing  the  influences,  natural,  political, 
social,  and  temperamental,  that   have   from   time  to   time   operated  in  the  de- 
velopment of  its  various  phases. 

Second. — It  exhibits  in  separate  chapters  such  biographical  and  historical 
matters  as  intimately 'concern  the  literary  lives  and  labors  of  the  acknowledged 
representative  writers  of  our  country  ;  together  with  standard  critical  opinions 
concerning  these  authors, — verifying  such  opinions  by  means  of  numerous 
interesting  and  characteristic  extracts  from  their  works.       Retail  Price,  $1.75. 


Berard's    History   of   the    United    States. 

NE\A/    EDITION. 
This  book  is  a  skillful  condensation,  not  a  mere  compilation, — written  in 
an  attradlive  and  pleasant  style,  which  cannot   fail  to  instnid);  and  interest  the 
learner.     Its  wide  sale  and  increasing  popularity  attest  its  merit. 

Retail  Price,  $1.20. 
.-•-. 

Leach's  Complete  Spelling  Book. 

Containing  a  systematic  arrangement  and  classification  of  the  difficulties  in 
Orthography,  arising  from  the  irregular  sounds  of  the  letters.  The  Best 
Speller  extant  for  Advanced  Classes.  Retail  Price,  32  Cts. 


These  books  are  all  Fresh,  Original,  Thoroughly  up  to  the  Times, 
and  especially  Adapted  to  the  Improved  Methods  of  Instruction 
which  now  prevail  in  the  best  schools.  They  are  in  very  general  use  in  all 
parts  of  the  United  States,  and  we  are  proud  of  the  fact,  which  has  been  so 
often  stated  to  us,  that 

They  are  Best  Liked  by  the  Best  Teachers. 

Upon  the  Liberal  Terms  offered  for  First  Introduction,  to  intro- 
duce these  books  will  in  most  cases  be 

MORE    ECONOMICAL 

than  to  continue  to  use  the  old  ones,  which  must,  in  many  instances,  soon  be 
replaced  by  new  books  at  full  prices.  Correspondence  is  solicited  with  refer- 
ence to  the  use  of  these  publications  in  public  or  private  schools. 

Copies  for  Examination  or  for  First  Introduction,  in  Exchange  for  other 
books  in  use,  will  be  supplied  at  HALF  RETAIL  PRICES,  on  application 
to  the  Publishers,  or  to  any  of  their  Agents. 

COWPERTHWAIT  &  CO.,  Educational  Publishers, 

628  and  630  Chestnut  Street,  Philadelphia. 


B@^  New  Illustrated  Descriptive  Catalogue  sent  free. 


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RETJLII.    I"«ICE    LIST. 

GEOGRAPHIES. 

Mairen\  New  Primary  Geography, 

New  Cummon  School  GVography, 
>'ew  Physical  Geography, 

Apgar's  Geographical  Drawing-Book, 

The  Geographical  Question-Book,    .  . 

Warren's  Physical  and  Outline  Charts,     • 

Political  ;ad  Outline  Charts'.         .       . 
Map-Drawing  Paper, 


$0.75 
1.S8 
1.88 

94 


(Per  5«-. 

(Per  Set,^     lu.UU 
.25 


GPAMMAKS. 

Greene's  Xew  Iiitroduetiou  to  English  Gramuiar, 
Xew  English  Giamiuar. 
Analysis  of  the  English  Language, 

ARITHMETICS. 

Hagar's  Primary  Lessons  in  Numbers, 
Elementary  Arithmetic, 
Common  School  Arithnutic. 
l»i(;atiou  Prohlems  an«l  Kev. 


.J6 

1.05 

.80 


.30 

•  .50 
l.uo 
l.uu 


READERS. 

Monroe's  Fi.st  RcaJer, 

Seo.nd  Keader,    .... 

Third  Keader, 

Eoiirth  Reader,  .       *       . 

Fifth  Reader, 

Sixth  Keader, 

H1S":^0RIES. 

Goodrich's  Chil.rs  History  of  the  United  States 
Berard's  School  History  of  the  Lnitcd  States,    '. 

MISCELLANEOUS 
Monroe's  Manual  of  Physical  and  Vnca!  Training- 
Royse's  Manual  of  Americau  T,itc-rat.  re, 
Berard's  Manual  of  Spanish  Art  and  Literature,  ■ 
Angdl  s  Manual  of  French  Literature 
Leach-s  Complete  Spelling  Bonk, 
Cowdery's  Moral  Lessons, 


(In  Preparation.) 


(New  Edition,) 


.30 

.fO 

.75 

1.00 

1.25 

1.50 


.50 
1.20 


1,00 
1.75 

■.'4 
1.1. 

.94 


New  Illustrated  Catalogue  Sent  Free, 
COWPERTHWAIT  &  CO. 


«-',V  .1    O.IO  ih,.s1.,ut  Sirevt, 

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